The Plateau Problem, Part I: Historical Survey (Studies in the Development of Modern Mathematics)

N

54

THE PLATEAU PROBLEM: PART ONE

(a)

(b)

Spheres with handles

Spheres with cross-caps

..

.

.. , ~

.

:

.•••

, .

;

'

I

1\

"

.

~.

...

...•.

"

Figure 36

;

-

(J

the angle formed by the radius vector with the vertical axis z. As

to ± ;,

(J

tends

IQ> I~oo and ~-+O, which specifies the behaviour of the two projections

(see Fig. 37). A direct computation shows that this curve is not the boundary of any finite-area surface immersed in R3. Moreover, there exist examples of such exotic curves, each point of which possesses a similar complicated structure and, therefore, the reduction of any arbitrarily small part of the boundary requires the construction of an infinite-area surface. We dwell on the properties of minimal surfaces whose existence is asserted by the Douglas theorem. Since smooth mappings of a surface in R3 that are immersions at all points except possibly a set of measure zero are being considered, minimal films can have self-intersections and branch points. It is clear that there exist contours for which there is no minimal film in the embedding class. For example, the

THEORY OF MINIMAL SURFACES

55

Figure 37

simplest knotted contour (trifolium) is shown in Fig. 38. There is no disc embedded into R3 that can span it. At the same time, there exists a smooth mapping of the disc D2 in R3 (with self-intersections) that realizes the minimum area in the class of mappings of the disc with given boundary. This is true, by the way, for any Jordan closed contour homeomorphic to a circle. In Fig. 38, an example of a minimal film spanning a circle embedded into R3 and knotted as a trifolium is also shown. This surface not only has self-intersections (three concurrent line segments), but also a branch point (in the centre). However, this surface is not a solution of the Plateau problem, that is, its area is not absolutely minimal in the class of all discs. An absolutely minimal surface (with analytic boundary) cannot possess branch points, and is always a smoothly immersed surface. For a more detailed discussion of the "surface", see below.

3. Singular points of minimal surfaces and three Plateau principles. An important property of minimal surfaces of general form is that they often have singular points, that is, points near which the surfaces are arranged in a more complicated way than the usual plane disc of small radius. The possession of singularities is a "typical situation" in the sense that, for an arbitrary boundary frame, the soap film spanning it "most probably" has singularities. Singularity sets can be fairly complicated and interesting. Rather often, singular

56

THE PLATEAU PROBLEM: PART ONE

Figure 38 points are not isolated and fill entire line segments where several sheets of a film meet. What is the character of singular points? Three experimental facts which we will call Plateau principles were discovered by Plateau himself. SECOND PLATEAU PRINCIPLE. Minimal surfaces (of zero mean curvature) and systems of soap-bubbles (of constant positive mean curvature) consist of fragments of smooth surfaces meeting along smooth arcs. THIRD PLATEAU PRINCIPLE. Portions of smooth surfaces consisting of films of constant mean curvature can meet only in the following two ways: (a) three smooth sheets meet on one smooth curve, (b) six smooth sheets (along with four singular curves) are concurrent at one point, a vertex. FOURTH PLATEAU PRINCIPLE. In the case when three sheets of a surface meet in a common arc, they form equal angles of 120 0 with each other. In the case when four singular curves (and six sheets) are concurrent at one vertex, these edges form equal angles of approximately 109 0 • The corresponding geometric diagrams are given in Fig. 39. Each of the soap films depicted is realized as a surface spanning the corresponding contour, since these films are stable. It is useful to picture one of the mechanisms of the

THEORY OF MINIMAL SURFACES

57

Figure 39

formation of three sheets of a surface, meeting in a common edge at the angles of 120 0 • Consider a frame inside a large soap-bubble (Fig. 40). Gradually releasing the air from the bubble, we see that it envelops the contour, and finally, collapses onto the surface of the required shape. We will say that a point of a minimal surface is of multiplicity k ifk sheets of the surface meet in it. A regular point has multiplicity two, since two sheets meet in a common edge at an angle of 180 0 , that is, the surface is smooth in a neighbourhood of the given point. It can be seen from Fig. 41 that two sheets of a minimal surface cannot meet in a common edge at an angle other than 71": otherwise, there is a film deformation decreasing its area. If we also consider surfaces with singularities, then, from the mathematical point of view, there exist films whose mean curvatures are zero almost everywhere (at non-singular points), and the multiplicity of the singular points can be greater than three. For example, two orthogonal planes meeting in a straight line realize such a surface. The multiplicity of each self-intersection point is here equal to 4 (Fig. 41). Nevertheless, attempts to construct a real soap film with the number of sheets meeting in the singular line segment greater than three fail. As it happens, any surface of zero curvature with self-intersections is unstable, that is, cannot be realized as a stable physical soap film. This statement even has a local character, rather than global: an arbitrarily small neighbourhood of each self-intersection point is already unstable. In fact, suppose that four

58

THE PLATEAU PROBLEM: PART ONE

Figure 40

Figure 41

THEORY OF MINIMAL SURFACES

59

sheets have met in some self-intersection interval of the surface. We can assume that two intersecting parts of the surface are mutually orthogonal: otherwise, the instability of the film is evident. Cut out a part of the film placed near the selfintersection interval, using the first Plateau principle, and fix the boundary of the two thin strips of the film so obtained. Consider a section of the film by a plane orthogonal to the singular line segment. The quadruple singular point decomposes into the union of two triple points, since the length of the curve originally consisting of two orthogonal line segments is greater than the length of each of the curves shown in Fig. 41 (verify!). Minimizing the length of the curve, that is, the plane section, we minimize the area of the surface portion distinguished. In Fig. 41, this process is shown in R3 in the immediate vicinity of the section. Since the film is now in stable equilibrium (in particular, small perturbations only increase its area), the resultant of forces acting at each point of the film equals zero. The same applies to singular points. In the equilibrium position, exactly three such sheets meet at each of such points; therefore, it is clear that they meet at equal angles 27r/3. Similar reasoning can be performed also when more than four sheets meet in a common edge. The surface gets restructured again, and occupies a position oflesser area (energy), as a result of which, a singular edge of high multiplicity decomposes into several triple edges (Fig. 41). Recall that, generally speaking, several soap films can span one frame. It is shown in Fig. 41 how singular points of large multiplicity can decompose into the union of triple points in several ways. We have considered above a section of a surface by a plane orthogonal to a singular edge. The deformation constructed was uniform at all points of the edge near to the section, that is, this deformation shifts the boundary of the orthogonal strips of the film (Fig. 41). However, it is easy to construct a deformation decreasing the area of the film also when the boundary of a portion of the film is fixed rigidly (see Fig. 41). The quadruple edge decomposes into the union of two triple bent edges whose ends are fixed. Thus, we have constructed a monotonic contraction of the film, which decreases its area with the boundary fixed. Gluing this portion of the film into the original film of large size, we obtain the required contracting deformation. The same process can be demonstrated by considering a smooth immersion of the two-dimensional disc in R3 (Fig. 42). The narrow strip of the film, having risen upwards, orthogonally intersects the main plane of the film. An attempt to realize this surface as a real soap film reveals its instability at once. The quadruple self-intersection edge decomposes into the sum of two bent triple edges with common ends. These problems are related to the two-dimensional Steiner problem, namely, how to find a thread of smallest length, connecting a finite set of points on the plane. The thread is allowed to ramify and have points of multiplicity three or more. By means of soap films, this problem can be solved experimentally. Take two close parallel planes made of thin organic glass, and join them with vertical column-segments placed at those points of the plane through which the required

60

THE PLATEAU PROBLEM: PART ONE

Figure 42 thread has to pass. Dipping the arrangement into soap water and removing it, we obtain a film spanning the columns and realizing the minimum of length of its plane section, the thread containing points of multiplicity not greater than three. Similar mechanisms act also in those cases where air pressure plays a part in forming a soap film, for example, when bubbles bounding a closed volume filled with air are formed in it. In this case, the film occupies a position opposing the pressure of the air inside. In the general case, by combining the spanning of the contour by a soap film with the process of blowing bubbles, we can obtain real films containing both bubbles and surfaces of zero curvature. These effects are observed in whipping soap foam. In Fig. 43, quite complicated boundary contours and some of minimal films spanning them, and constructed by Almgren and Taylor are shown l2 . The interaction of different surface sheets can often be seen to be intricate, and get complicated with more complex boundary contours. Sometimes, soap films with bubbles are found spontaneously in removing sufficiently complex contours from liquids.

4. The realization of minimal surfaces in animate nature. As early as the eighteenth century, it was known that many concrete problems of physics, chemistry, biology reduce to the analysis of minimal surfaces. We give

THEORY OF MINIMAL SURFACES

61

Figure 43

an example from biology. Minimal films happen to be widely spread in nature as most economical surfaces forming the skeletons of some living organisms. The most effective example is given by the remains of radiolarians, tiny marine organisms having most various and exotic forms. Apparently, D.A.W. Thompson was first to pay considerable attention to the fact that surface tension played an essential part in forming these living things in his book "On Growth and Form". Radiolarians consist of small lumps of protoplasm in froth-like forms similar to soap bubbles and soap foam. Since these organisms are rather complex, the minimal surfaces in their structure, generally speaking, have many branch points and triple singular edges, on which most of the mass of the liquid in the composition of the organism is concentrated. This concentration along singular edges and near singular points is clearly seen in real soap films. The liquid flows from the film freely until an edge is encountered where the sheets of the film meet. Here, the liquid slows down and subsides, forming water line segments which make the singular edges of the minimal surface visible. The same process actually takes place in radiolarians. The concentration of the liquid along the branchings leads to solid particles in seawater and salt settling along the edges, and gradually forming the hard skeleton of the animal. The geometry of this skeleton can be represented visually if, in a mass of soap foam, the system of singular edges along which different bubbles meet is considered. These line segments are joined sometimes in quite an intricate way, and form a "liquid skeleton" of the foam. After the death of the animal, the soft tissues gradually vanish, decompose, and a small solid skeleton formed in the indicated way is left. It is a visual still image ("photograph") of the

62

THE PLATEAU PROBLEM: PART ONE

system of the branchings arising from the complex minimal surface with the bubbles included. Furthermore, in the formation of the solid skeleton of radiolarians, a definite part is played by the walls of the cells, which also accumulate sea-salts. In Fig. 44, three skeletons of radiolarians are shown. The picture was borrowed from the book by Ernst Haeckel "Report on the Scientific Results of the Voyage of the HMS Challenger during the Years 1873-1876". There are also depicted three minimal surfaces (with soap-bubble inclusions) which span contours making up the edge systems of an ordinary tetrahedron, cube and prism. The similarity of the forms of real organisms to the indicated films is striking. The skeletons of radiolarians are seen to reproduce the branching pattern of minimal surfaces quite accurately. Another two exotic radiolarians' skeletons are represented in Fig. 45. And this is by no means a unique example of the role played by minimal surfaces in animate nature surrounding us. The role of such films is also important in chemistry, where surface interactions on the surfaces of separation of media determine the character and velocity of chemical transformations. Examples of minimal surfaces are given by the well-known membranes found in natural formations and some fields of human activity. Such are the ear-drum or membranes between living cells, etc. The principal property of a membrane is the tendency to occupy a position optimal from the energetic point of view for a given environment. Therefore, the shape of a membrane is to a considerable degree determined by external influence and, conversely, from the form of a minimal surface, it is possible to make a judgement concerning the interactions optimal for the organism. Real membranes possess some other important properties: selective permeability or semi-permeability which enable the cells to exchange ions. We see that minimal surfaces are related to a number of interesting mathematical problems. Their correct definition and investigation allow us to understand this wonderful, though outwardly simple, natural phenomenon better. Other details see in: [8], [12], [18], [34], [35], [36], [48], [49], [53], [56], [105], [116], [117], [135], [136], [139], [140], [141], [143], [144], [145], [146], [147], [189], [194], [198], [205], [207] - [210], [215], [217], [218], [249], [250], [260], [278].

THEORY OF MINIMAL SURFACES

(a)

(b)

(e)

a.'

10'

c'

Figure 44

63

64

THE PLATEAU PROBLEM: PART ONE

(a)

Figure 45

D SURVEY OF SOME IMPORT ANT PUBLICATIONS IN MINIMAL SURFACE THEORY FROM THE NINETEENTH TO THE EARLY TWENTIETH CENTURY §1. Monge (1746-1818). In this chapter, we give some important fragments of the works of Monge, Poisson, Plateau, Rad6, and Douglas, which had a strong influence on the whole of the further development of the theory of minimal surfaces and variational calculus in the large. We start with the well-known paper of Monge [123],1, which laid the foundation for the analytic and geometric theories of the twodimensional minimal surfaces. Of particular interest to us is §XX of his work, which we reproduce here in its entirety. Gaspard Monge

Application de l'analyse Ii la geometrie.

§xx On the curved surface whose two radii of curvature are always equal to each other and of opposite signs.

I.

We have seen that the common expression of the two radii of curvature at the same point of a curved surface is

In order that the two values of R given by this expression be different only in sign, it is required that h = 0; therefore, substituting this value for h, the second partial differential equation of the required surface is (a)

J

(1 + q2)r - 2pqs + (1 + p2)t =

Translated from the French (tr.).

o.

66

THE PLATEAU PROBLEM: PART ONE

This equation is the same as that found by M. Lagrange for the surface with minimal area; therefore, the surface which we consider enjoys another remarkable property, namely, if we encircle a part of its area by some contour, continuous or discontinuous, then, of all the surfaces which pass through this contour, it is the one whose area inside the contour is the least.

II. To integrate equation (a), we shall first find the equations of the characteristic of the surface to which this equation belongs, and for that we shall differentiate, regarding r,s,t as unique variables, which will give:

R'

=1+

q2, S'

=-

2pq, l'

= 1 + p2;

substituting these values in

R'dy. - S'dx dy + l' dx 2 = 0, we have the first equation of the characteristic, (b)

(1 + q2)dy. + 2pq dx dy + (J + p2)dx 2 = 0.

Then, eliminating r,s,t from the proposed equation, and substituting the value which equation (b) gives for : ' we have for the second equation of the characteristic: (c)

(1 + q2) dp2 - 2pq dp dq + (J + p2) dp2 = O.

This latter ordinary partial differential equation between the two variables p,q is easy to integrate; for, if we integrate it, regarding q as the principal variable, we find ddp

= 0,

the integral of which is dp = adq, a being our arbitrary constant introduced by the integration. Substituting the value of: given by this integral for h, the integral of equation (c) will become

SURVEY OF MINIMAL SURFACE THEORY

(1 + q2) a 2

-

67

2apq + 1 + p2 = 0,

or

1 + a 2 + (p - aq)2 - 0. This equation has two roots; thus, the surface has two distinct characteristics, so that if we represent by a our arbitrary constant of the first characteristic, and by {3 that of the second, the distinct equations of these two curves will be

p - {3q = - -J - 1 - {32, a + {3 =

which gives

2pq

1 + q2'

1

a{3 =

a =

+ p2

T+q2'

pq+-J-p2_q2 1 + q2 '

~------''-::-~'-

_pq--J-l-p2-q2 l+q2 '

{3-

and hence, determining the values or p and q, we have: (a - (3) p = - {3-J - 1 - a 2 - a-J - 1 - {32, (a - (3) q = - -J - 1 - a 2 - -J - 1 - {32.

As to the other equation of each of the two characteristics, it is included in equation (b) which can be put into the simple form dx 2 + dy2 + dz2

= 0;

and which, being the sum of three squares, produces the three equations dx = 0, dy = 0, dz = 0,

of which, because dz = pdx + qdy, any two entail the third, and which, consequently, can be reduced to two of them, e.g., the first two dx = 0, dy=O.

68

THE PLATEAU PROBLEM: PART ONE

If these two equations which follow from (b) were factors of this latter, then each of them could take place separately, and they would belong, one to the first characteristic, and the other to the second; but equation (b) produces them both simultaneously: they are therefore both valid for each characteristic. Hence, each of these curves is not determined by only two equations, as in all the cases which we have considered so far, but by three equations; therefore, finally, each characteristic reduces to a point. Thus, the analysis does not present this surface to us, as generated by the movement of a generator variable in figure and position, by virtue of the variation of one of these parameters; it shows it as generated by the movement of a variable point by virtue of the variation of two of the constants which, at each moment determine its position.

Otherwise. If we consider the required surface as the envelope of the space, on which a moving surface changeable in figure varies, then the envelope can move along two different directions. For each of these directions, two consecutive envelopes cut each other in a curved line, and this is the point of intersection of these two curves which is in the surface. There is therefore this difference between the surface of §XIX and the one under consideration, which consists, for the first, in the possibility for the point of contact to move, at each instant, only along one direction, and consequently, to generate in the whole course of its movement only a curved line or a surface whose area is zero, whereas, for the actual surface, the generating point, able to move at each moment in two different directions, generates a curved surface whose area is not zero. It follows from this that, for the first characteristic, the three quantities a, x, y are all three constant; thus, the first of them is a function of the other two. It is the same with (3 considered in the other characteristic. Therefore, in general, the two coordinates x, y, and, hence, the third one, z, are functions of a and (3. It only remains to determine the forms of these three functions so that the proposed equation may be satisfied; but this consideration which, in addition, would lead us to the same result, would deviate us from the path of integration that we are going to take.

III. Though equation (b) produces two other simultaneous equations, we can treat it as a unique equation. By substituting the values of p and q, it becomes dy2 + (a + 13) dx dy + a(3 dx 2 = 0,

or (dy + a dx) (dy + (3 dx) = 0;

69

SURVEY OF MINIMAL SURFACE THEORY

which, being composed of two factors, gives, for the two characteristics, the two equations dy + a dx = 0, dy +

~

dx = 0;

but, in these two equations, it is the second which belongs to the first characteristic, and the first which belongs to the second. In fact, if, after elminating r,s,t from the proposed equation, which gives (1 + q2)dp dy + (1 + p2)dq dx = 0,

we

substitute

of a,

for

dp the value dq first characteristic, then we have

which

corresponds

to

the

(1 + q2) a dy + (1 + p2) dx=O,

whence dy +

~

dx=O

Also, employing for dp the value of dq second characteristic, we find dy + a dx

13,

which corresponds to the

= O.

The two equations of the first characteristic are therefore

p - aq = dy +

-J -

~dx =

1 - a 2,

0,

and those of the second

p - ~q

=

-J -

1 - ~~

dy + a dx = O.

If ~ were constant also for the first, for which we have a = constant, then the second equation would be integrable, and its integral would be completed by an arbitrary constant which would be a function of a; but ~ is variable. If, therefore, we integrate, regarding ~ as constant, then the integral must be completed by a function of a and ~, and this function must be such that the differential equation

70

THE PLATEAU PROBLEM: PART ONE

on this integral, taken while regarding {3 as the unique variable, should be satisfied; this will not yet be sufficient for its determination. Representing therefore by q> this function of the two quantities a, {3, the integral of the second equation will be at first (d)

y

+ (3x x

(e)

= q>(a, (3),

= q>".

For the same reason, if we represent by \jI another function of a and (3, then the integral of the second equation of the other characteristic will first be given by the two equations (f)

y + ax = 1/;{a,(3),

(g)

x = 1/;',

But the point of the surface must be determined by the intersection of the two characteristics and, hence, four equations (d), (e), (f), (g) must be valid for its projection onto the plane of x and y, and they cannot be valid unless the functions q> and \jI satisfy the following two: (h)

q> - {3q>" = \jI - a\jl',

(i)

q>"

= \jI',

which result from the elimination ofx and y, and whose integrals must serve to determine the forms of the functions q> and \jI. To integrate these latter two equations, it is first required to separate the functions, and for that, we shall differentiate them, regarding successively a, then {3, as unique variables; and, first eliminating all that depends on one of these functions, and then all that depends on the other, we shall find the two second partial differential equations. (a - (3) q>" 1 + q>' (a - (3)

\jI" 1

+

= 0,

\jI" =

0,

which can be transformed into the following q>'

d--

a-{3 = 0, d{3

\jI"

d--

a-{3 = 0, da

SURVEY OF MINIMAL SURFACE THEORY

71

and whose integrals are ' = (a - (3)" a, \II"

= (a - (3)'it" {3,

in which and 'it are two new arbitrary functions of a unique quantity, a for one, and {3 for the other, and which we accentuate because of the subsequent integrations. These two equations are themselves exact differentials taken, one while making only a vary, the other while making only {3 vary; and their integrals are = (a - (3)'a + a + F{3, = (a - (3)'it' {3 + 'it{3 + fa,

in which f and F are two new arbitrary functions of a unique quantity. Of the four arbitrary functions which are involved in these equations, only two are necessary, because it is voluntarily that we have differentiated two equations (h), (i), and that we have come to differentials of the second order. It is therefore required to determine the forms of two of these functions so that equations (h), (i) should be satisfied. Thus, we find from these integrals by differentiation that " = <1>' + F',

1/;' = 'it' + /" and, substituting for , ", \II, \II' their values in (h) and (i), we find that they are satisfied provided that F{3 = 'it{3, fa = - a,

which determines the forms of the two supernumerary functions F and f. Hence, substituting the expressions for these functions in the equations for , \II, ", \II', we have: = (a - (3) <1>' a - a + 'it{3, \II = (a - (3) 'it'a - a

"

+ 'it{3,

= - <1>' a + 'it' {3,

\II' = - <1>' a

+ 'it' {3;

72

THE PLATEAU PROBLEM: PART ONE

finally, substituting these values in equations (d), (e) or in (f), (g), and finding the values of x, y, we have for the coordinates of the point of intersection of the projections of the two characteristics (1)

x = - <1»' a + v' {3,

(m)

y = -
Thus, replacing a and (3 by their values in terms of p,q, equations (1) and (m) give, in terms of p and q, the values for x, y which we should obtain from the first two integrals of the proposed equation, one of these integrals being completed by the arbitrary function <1», the other by v; and we should obtain these two first integrals if we could derive two other equations from (1) and (m), each of which would contain only one of these functions and its derivatives. The two equations (1) and (m) are not therefore the first integrals of the proposed equation; but, taken together, they express the same thing as the two first integrals also taken together; and we shall see that they lead to the finite integral just as directly. IV. In fact, if we differentiate the two equations (1), (m) so as to have the values of dx, dy, and if we substitute these values, as well as those of p, q, in dz = pdx + pdy, then we find

an equation in which the variables z, a, {3 are separated, and which gives

the second member of which, when the functions <1», V are determined, will depend only on quadratures. Therefore, the coordinates of each point of the surface are given in terms of a and {3 by the three equations (1), (m), (n); hence, the equation of this surface in x, y, z, or the finite integral of the proposed equation is the result of the elimination of the two indeterminates a and {3 from these three equations.

V. The point is now to construct this integral, or, what is the same thing, to find the generation of the surface. The unique construction to which we have now come proceeds by means of infinitely close curves: it proves, in fact, that the area

SURVEY OF MINIMAL SURFACE THEORY

73

of the surface is not zero, which, by the way, is evident; but it is of no use in practice. Weare nevertheless going to give it, because it may give rise to more successful efforts. Let a double-curvature curve be arbitrarily taken in space. Assume that one of its tangents moves constantly touching it; it will generate a developable surface, for which the arbitrary curve will be the edge of regression; and a certain point of the tangent will describe a curve to which the moving tangent will be constantly normal, and which will be the evolute of the arbitrary curve: we shall see that this evolute will be one of the lines of curvature of the required surface. Produce each of the radii of the evolute beyond this curve by a quantity equal to itself; this will determine on each of the tangents to the arbitrary curve, a point which we shall call second centre, because it will be the centre of the second curvature of the corresponding point of the surface, the point whose centre of the first curvature will be the point of contact of the arbitrary curve. Imagine that a circle of variable radius moves so that I ° its plane always passes through the moving tangent and is always normal to the developable surface described by this tangent; 2° its centre at each moment coincides with the second centre; 3° the circumference always passes through the corresponding point of the evolute and, therefore, intersects it at right angles; the circumference of this circle will generate a curved surface which will pass through the evolute in which this latter line will be a line of curvature and whose two radii of curvature will be equal to each other and of different signs for each point taken on the evolute. We shall have to consider only one infinitely narrow zone in this surface, the zone bounded by the evolute. We then suppose that each tangent can move around the second centre through which it passes, and in the plane of the corresponding circle, i.e., in the plane normal to the developable surface; then, let one ofits tangents in fact move and describe an infinitely small and arbitrary angle around the second centre; let the next tangent also move around its second centre until it intersects the previous tangent considered in its new position; let the third tangent move also until it intersects the second, and so on and so forth for all the tangents. All these straight lines considered in their new positions will still be normal to the curved surface generated by the circumference of circle; and because they consecutively meet pairwise, they will be in a second developable surface whose edge of regression will be distinct from the first arbitrary curve, but infinitely close to it. The second developable surface will be normal to the surface generated by the circumference of the circle; it will intersect it in a curve which will be the next line of curvature, and which will be an evolute of the edge of regression of the second developable surface. The zone included between this second evolute and the first will belong to the required surface. Now, if we operate with the edge of regression of the second developable surface and on its evolute, as we have operated on the arbitrary curve and on its evolute, then we shall have the third line of curvature of the required surface;

74

THE PLATEAU PROBLEM: PART ONE

and, continuing thus further, we obtain all the points of the surface. This construction has all the necessary generality; for it assumes a first curve to be entirely arbitrary, and, consequently, an arbitrary function for each of the projections of this curve. We shall not give any more examples of a surface whose generation can be expressed by equations in partial differential of the second order. As to those which require differential of the third order, we shall only consider the small number of those which are remarkable by their simplicity and use in applications.

§2. Poisson (1781-1840). In this section, we turn to the work of Poisson. We make use of the excellent account of Poisson's investigations performed in 176 • In that survey, the works of Poisson are not only quoted, but also supplied with detailed and expert comments. We have therefore decided to retain these comments as essentially completing and clarifying sometimes rather bulky texts by Poisson himself (see l76, pp. 75-80 and 94-99).

Isaac Todhunter A history of the progress of the Calculus of Variations during the nineteenth century Chapter IV 103. The twentieth section contains some reductions of the variation of a double integral. Consider the definite integral U = HVdxdy.

By the known rules for the transformation of double integrals, if we consider x and y as functions of two other variables u and v, we must put dx dy = ( -dX -dy - -dx -dY ) du dv· du dv dv du '

so that we have U =

Now put x + ox, Y + oy,

Hvldx

dy _ dx dY..\du dv; Ifu dv dv dU)

Z

+ oz in place of x, y,

Z

under the integral sign.

SURVEY OF MINIMAL SURFACE THEORY

75

From what was said above it will be sufficient that OX, oy, OZ should be arbitrary functions ofu and v, and it will not be necessary to vary the limits relative to u and v in order that the integral U may vary in the most general manner both with respect to the limits relative to x and y, and with respect to the form of the function. The complete variation of U will then be

d~\oVdu dv

OU = JldX dy _ dx 'lfu dv dv dU)

+ JldX doy _ dx doyl + dy dox _ dy dOx.\ Vdudv. \du dv

dv du

dv du

du dV)

But by the formulae of the preceding article we have dx doy _ dx doy = (dX dy _ dx d~\doy du dv dv du du dv dv dU) dy' dy dox _ dy dox = (dX dy _ dx dy.\ dox. dv du du dv du dv dv duJ dx Hence oU = H(oV + V dox + \ dx

v dOY) dy

f.dx dy _ dx dY.\du dv, \,du dv dv duJ

that is, by restoring the variables x and y OU =

H( 0V +

V -dox + V -dOY) dx dy. dx dy

In this formula the limits are the same as those ofU. Now substitute the value of OV given by equation (2), and observe that V' ox + V dox dx

=

d. Vox dx'

V,oy + V doy = d. Voy. dy dy' thus oU

=

[JVox dy] - (JVox dy)

+ [JVoy dx] - (JVoy dx) + JS(Nw + Pw' + Qw, + Rw" + Sw,' + Tw" + ... ) dx dy ..... (3)

76

THE PLATEAU PROBLEM: PART ONE

By the method of integration by parts we may remove the differential coefficients of w from under the double integral sign. For HPw'dx dy = UPw dy] - (JPw dy) HQw,dx dy = UQw dx] - ([Qw dx)

HP' 10 dx dy, - HQ 10 dx dy.

By two successive integrations we have HRw" dx dy = [JRw'dy] - (JRw'dy) - [JR'w dy] + (JR'w dy) + HR"w dx dy, HTw"dx dy = UTw,dx] - (JTw,dx) - UT,w dx] + (JT,w dx) + HT"w dx dy.

By integrating first with respect to y and then with respect to x we obtain HSw,'dy = USw'dx] - (JSw'dx) - US,w dy] + (JS,w dy) + HS,'w dx dy;

by performing the integrations in the reverse order, we obtain HSw,'dx dy = USw,dy] - (JSw,dy) - [JSw dx] + (JSw dx) + HS,'w dx dy;

For the sake of symmetry we may use the half sum of these equivalent expressions, that is 1 1 1 1 HSw,'dx dy = - [JSw'dx] + - [JSw,dy] - - (JSw'dx) - - (JSw,dy)

2

1

- 2"

222

[JS,w dy] -

111 2" [JSw dx] + 2" (JS,w dy) + 2" (Sw dx)

+ HS,'w dx dy. The subsequent terms in the last part of the formula (3) may be transformed in a similar manner. Thus the expression for 5U will become finally

5U

=r +

HH

10

dx dy ....... .

where for shortness, H = N - P' - Q + R" + S,' + T" - ...

(41),

SURVEY OF MINIMAL SURFACE THEORY

r

= [JVox dy] - (JVox dy)

77

+ [JVoy dx] - (JVoy dx) 1

1

+ [JPw dy + JQw dx - fR'wdy - 2" JS,wdy - 2" fSwdx - JT,wdx + ... ] - (JPw dy + JQwdx - JR'wdy -

+ [JRw'dy +

~ fSw'dx

1

1

2" JS,wdy - 2" JSwdx - JT,wdx + ... )

+ fTw,dx +

~ JSw,dy

- ... ]

1 1 - (JRw'dy + - JSw'dx + JTw,dx + - JSw,dy - ... ) 2

2

+ ........................ . The two expressions which have been found for HSw,' dxdy must be identically equal; hence we have [JSw'dx] - (JSw'dx) - [JS,w dy] + (JS,w dy) = [JSw,dy] - (JSw,dy) - [JSw dx]

+ (JSw dx).

This may be written [J(Sw' + Sw) dx - (f(Sw' + Sw) dx) = [J(Sw,

+ S,w) dy] - (f(Sw, + S,w) dy) ............ .

(5).

This will be verified presently (see Art. 106). We may observe here that Sw' + S'w is the partial differential coefficient of Sw with respect to x before substituting the value of y obtained from one of the limiting equations. But the value of(Sw' + S'w) dx after we substitute for y its value is no longer a complete differential with respect to x and thus cannot be integrated immediately. Similar remarks apply to the term (Sw, + S,w) dy. [This remark guards against the error indicated in Art. 27.] 104. The twenty-first section. For U to be a maximum or minimum we must have OU = 0. The double integral included in equation (1) cannot be reduced to simple integrals because w is an arbitrary function of x and y; it will therefore be necessary that the two parts of this formula should separately vanish. Thus we obtain

r = 0,

H

= 0,

for the equations which must be satisfied in order that the double integral which we are considering should have a maximum or minimum value. The second

78

THE PLATEAU PROBLEM: PART ONE

equation will serve to find z in terms of x and y; this equation will be in general a partial differential equation of the order 2n if V be of the order n. The first equation will decompose into others the number and nature of which in the different cases which may occur we will investigate in the subsequent articles. This is the most delicate part of the question. The preceding analysis may be extended without difficulty to triple and quadruple integrals, &c. In the case of a triple integral, for example, we shall obtain for the variation an expression analogous to that in equation (4); this expression will consist of a triple integral, and of another part containing only double integrals which relate to the limits of the triple integral we are considering. We might also suppose that the quantity under the triple integral sign involves unknown functions of the independent variables, and that these functions are independent, or that they are connected by given partial differential equations. We shall not stop to consider these questions, since they present no new difficulties and no useful applications. The determination of the relative maxima or minima of multiple integrals can be reduced to the determination of absolute maxima or minima by the method of the thirteenth section, which is obviously applicable whatever may be the number of independent variables. Thus, for example, if the first of the double integrals HVdx dy, HTdx dy, HWdx dy, ...

is to be a maximum or minimum, and at the same time the others are to have given values, the problem amounts to investigating the absolute maximum or minimum value of

H(V + aT + bW

+ &c) dx dy;

where a, b, ... are unknown constants which must be determined by means of the given values of the integrals. We suppose here that these integrals and the first integral are all taken between the same limits. 105. The twenty-second section. [The results from this point to the end of the memoir were not known before the publication of the memoir.] Let us now examine the equations relative to the limits of U which are necessary for the maximum or minimum of this double integral, and which must be deduced from the condition r = O. In order to render the reasoning easier to follow, we will suppose that x, y, z are the rectangular co-ordinates of any point of the surface determined by H = 0, and that the integral U corresponds to a zone of this surface comprised between two closed curves which will be given or which will have to be determined. Let ABC be the projection of the exterior curve upon the plane of (x,y), and DEF

SURVEY OF MINIMAL SURFACE THEORY

79

that of the interior curve upon the same plane. The integral relative to x and y which

aU represents will extend to all the elements of dx dy of the plane area intercepted between these two curves. It may however also be considered to represent the excess of a double integral extended to all the elements of area enclosed by the curve ABC over the same double integral extended to all the elements of area enclosed by the curve DEF. Now aU reduces to r since by supposition H = 0; and r = r (\) - r (0) where r (\) denotes that part of aU which arises from the area bounded by ABC and liO) that which arises from the area bounded by DEF. Since these two limits ABC and DEF are in general independent of each other, the equation r = 0 will decompose into two others, namely,

r (\) = 0, r (0) = o. It will be sufficient to consider one of these; the other will be of the same form and susceptible of the same transformations. 116. In his twenty-eighth section Poisson makes some remarks on the mode in which the arbitrary functions are to be determined in some problems. There are particular problems in which the curve which forms the inner boundary of the required surface according to the hypothesis of the twenty-second section (Art. 105) does not exist, and in which consequently the conditions relative to this curve must be replaced by others, in order that the arbitrary functions which are involved in the general integral of the equation H = 0 may not remain undetermined, and that these problems may be completely solved. This circumstance might occur, for example, in the question where we have to find a surface of which the area should be a minimum between certain limits. The equation H = 0 is then a partial differential equation of the second order, and its integral involving two arbitrary functions is known in a finite form. Now if the minimum area is to be a zone included between two given curves, we see that these two curves through which the required surface is to pass will theoretically serve to determine the two arbitrary functions which occur in the equation to the surface, that is, in the integral of the equation H = 0, the only difficulty being that which arises from the complicated form of this integral. We see too that these two curves might be exchanged for other pairs of conditions. But if we require that the minimum area should be all that portion of the surface which is bounded by the exterior curve, it seems then that the integral of the equation H = 0 will have a greater degree of generality than the problem, and that the given curve will not be sufficient for the determination of the two arbitrary functions. 177. In order to remove this apparent indeterminateness, suppose we exchange the rectangular co-ordinates x and y for polar coordinates rand (J, where r is the radius vector and the angle which r makes with a fixed line drawn through the origin in the plane of (x, y). Put the origin within the boundary

80

THE PLATEAU PROBLEM: PART ONE

formed by the projection DEF of the interior curve (Art. 105) when such curve exists. Let r = f{0), z = cp(O), be the two equations of this curve; and

r = F(O), z = <1>(9), those of the exterior curve of which ABC is the projection. For the zone of minimum area the values of r will extend from r = f{0) to r = F(O), and those of 0 from 0 = 0 to 0 = 211", and the arbitrary functions which occur in the integral of H = 0 must be determined so that z should become cp(O) and «1>(0) for r = f{0) and r = F(O) respectively. Outside the zone, that is, for values of r less than f{0) or greater than F(O) whatever 0 may be, the ordinate z will be subject to no limitation and can become infinite. But if the minimum area is to be all that portion of the surface the projection of which is bounded by the curve ABC, the values of r will extend from r = 0 to r = F(O) for every value of 0, and throughout this extent the ordinate z must be finite. We shall therefore suppress in this case that portion of the integral of H = 0 which would become infinite when r = 0; and the integral thus modified will be reduced to the degree of generality which the problem has; so that the single condition that z should be equal to «1>(0) when r is equal to F(O) will suffice for completing the solution of the problem. Thus the solution of the question of the minimum area and of similar questions, separates into two problems which are quite distinct so far as relates to the determination of the arbitrary functions. I only here indicate this distinction which I will take up on another occasion. If the required surface is closed on all sides, so that for example we have to find the surface of greatest area which incloses a given volume, the conditions for this relative maximum will not furnish any equation suitable for determining the two arbitrary functions which the complete integral of the equation H = 0 when applied to this problem will involve. It is by means of other considerations that this integral must be reduced so as to contain only three arbitrary constants, namely the three co-ordinates of the centre of the sphere which solves the problem; the radius of the sphere will be determined by means of the given volume. I propose to consider this particular question in another memoir. [It does not appear that Poisson ever returned to the two problems which he proposed in the above section to consider at a future period.] 118. In the remaining three sections of the memoir Poisson discusses an example. In an addition to the work entitled Methodus inveniendi lineas .... Euler determines the figure of the elastic lamina, properly so called, by means of a principle communicated to him by Daniel Bernoulli, namely, that the integral ds 2 taken throughout the length of the curve should be less than for any p

SURVEY OF MINIMAL SURFACE THEORY

81

other curve of the same length; ds being the differential element of the sought curve and its radius of curvature. In order to give an example of the employment of the preceding formulae, we will extend this principle by induction to the figure of equilibrium of an elastic lamina which is curved in every direction and the points of which are not acted on by any given force. Thus denoting by e and ~ the two principal radii of curvature at any point of this surface, or more generally the radii of curvature of two normal sections at right angles, and by dO" the differential element of the surface, we shall suppose that among all surfaces of the same area the elastic surface is that which gives a minimum value to the

t)

integral

IS (~ +

integral

IS (~+

dO". [This is what Poisson says, but he really takes the

2

tY

dx dy; the two however coincide to the order of

approximation which he finally preserves.] 1 1 By the theory of the curvature of surfaces we know that the sum Q + ~ has the

same value for every pair of normal sections at right angles passing through the same point. With the notation already adopted, we have

+ Z,2) Z"__- 2z'z,z,'__+ (1 + z'2) Z" _1 + _1 = (1 __

e

~

~~

~

~~~

~~~-L~

(1+z'2+z,2)~

or, which is the same thing,

1

1

,

- + - = u +v e ~ "

where u =

z' z, v = -r.==;;===;< .Jl + z,2 + z2' .Jl + z,2 + z2' , ,

We have also do = .Jl + z,2 + z,2 dxdy.

Let c denote an undetermined constant, and put

v=

(u' + v.f + 2c.Jl + z,2 + z,2;

then the question amounts to making the integral IS V dx dy an absolute minimum. (see Art. 104.) The quality N of the nineteenth section (Art. 102) will be zero, and P, Q, R, S, T, will have for values P = 2 (u' + v,) (dU' + dV,) + 2cu, d'7.' dz'

82 Q

THE PLATEAU PROBLEM: PART ONE

= 2 (u' +

dV) v,) ( -dU' + ' + 2cv, dz, dz,

dV) R = 2 (u' + v) ( -dU' + ' ,

S

= 2 (u' +

dz"

dU' v) ( - ,

dz,'

dz'"

dv ) + -'dz,"

dU'- + dV) T = 2 (u' + v,) ( ' . dz"

dz"

It will be sufficient to substitute these values and their first and second differential coefficients with respect to x and y in the equation H = 0 of the twenty-first section (Art. 104), in order to obtain the indefinite equation to the elastic surface; this equation will be a partial differential equation of the fourth order. We must also substitute these same quantities in the equations of the twenty-sixth section, in order to obtain the equations relative to the perimeter of the elastic surface in all the cases which can occur. We will confine ourselves to writing these equations for the case where the elastic surface differs but little from a plane figure parallel to the plane of x and y; and we shall neglect consequently the terms in V of the fourth degree with respect to partial differential coefficients of z. Thus the values of P, Q, ... and therefore the equations in question will be exact as far as quantities of the third order. Thus we have, simply

v=

(z" + z,,)2 + 2c +

C

(z'2 + z,2),

from which we obtain

N

= 0, P = 2cz',

Q

= 2cz" R = T = 2 (z" + ZIt)' S = 0;

thus the equation H = 0 will become z"" + 2z;; + z"" - (z" + z,,) =

o.

If we denote by r a new variable, we may replace this equation by the following system of two equations of the second order: zIt

+

ZIt

=

r, f' + r" = cr .................... .

(a).

In consequence of these values ofR, S, T, the quantity z of the twenty-fourth section (Art. lO7) will be equal to 2r. In order to fix our ideas, I will suppose that the limits of the elastic surface in equilibrium are curves fixed and given, but that the tangent plane to this surface is not restricted by any condition throughout the

SURVEY OF MINIMAL SURFACE THEORY

83

perimeters of these curves; hence it will follow from the second case of the twenty-sixth section (Art. Ill), that we must combine with the two equations of each limiting curve the equation Z = 0 or r = 0, in order to form the two systems of simultaneous equations, which with the given area of the elastic lamina will serve to determine the constant c and the arbitrary functions contained in the integrals of equations (a). The area of the lamina cannot differ much from that of its projection on the plane of x and y; denote the area of the projection by A, and that of the lamina by A (1 + g) so that g is a very small positive fraction; we shall have A(1 + g) =

H-v'l + Z,2 + Z,2 dx dy,

or to that order of approximation which we have adopted Ag =

t

H(Z'2 +

Z,2)

dx dy .................... .

(b).

§3. Plateau (1801-1883). The present section is particularly important for this book. We give in it some fragments of the famous paper by Plateau 140, which we have already talked a great deal about in Chapter I of our book. It was in this work that Plateau laid the foundations of the study of the main qualitative effects which regulate the structure of two-dimensional minimal surfaces. It was in 140 that he described his most interesting experiments with soap films and gave a qualitative explanation for the phenomena of stability and instability of films. We quote the following fragments from 140: pp. 91-107 (§§55-65) and pp. 312-363 (§§179-203)1.

J.Plateau

Statique experimentale et thiorique des liquides soumis aux seules forces moliculaires. §§55. We now consider the figures which are formed when the rings are farther apart than required to obtain a cylindrical body. If, after having formed between the two rings a vertical cylinder whose height is much less than that corresponding to the stability limit, we lift the upper ring a little, then we see the cylinder shrink slightly in the meridional direction so as to form a waist; if the ring is lifted higher, then the waist becomes still more noticeable, while the figure is always perfectly symmetrical on both sides of the gorge circle which is situated, eventually, in the middle of the interval between the rings. If, in the cylinder from which we proceeded, the ratio between the height and diameter is of a suitable value, then we can, by continuing the process, make the waist very pronounced; and then the meridional 1

Translated from the French (tr.).

84

THE PLATEAU PROBLEM: PART ONE

line changes the sense of its curvature and shifts towards the rings, in such a way that two inflection points are situated at equal distances from the two sides of the gorge circle; furthermore, the bases of the figure preserve their convex shape, and even their curvature augments more or less. In this experiment, there is always, as can be imagined, a limit for moving the rings apart beyond which equilibrium is no longer possible; if it is surpassed, the waist spontaneously narrows and breaks; the figure separates into two portions; but, if we move the rings apart less than the limit in question, then the equilibrium is stable. The cylinder which seems to be able to represent the above results in the most pronounced manner is one with the ratio of the height to the diameter of approximately 5 to 7: for example, by employing rings of 70 mm in diameter, it is required to form a cylinder of about 50 mm height; the upper ring can then be lifted until it is at a distance of approximately 110 mm from the other, and we thus obtain a figure in which the gorge circle has diameter of only about 30 mm. The experiment performed in this way calls for great accuracy: the densities of the two liquids should be equal and the homogeneity of the oil perfect, while in approaching the limit for drawing the rings apart ones actions should be as careful as possible. But, proceeding so as to keep the axis of revolution horizontal, we can succeed without difficulty: the vertical 70 mm rings should be placed in advance at a distance of 110 mm from each other, each fixed with its lower part to a vertical iron wire, while also fixing the lower extremities of the wires to an iron plate which supports the whole arrangement; finally, the wires themselves should be covered with cotton so that the oil cannot cling (§9)1. First, a cylinder is formed between the two rings, and then the volume of the mass is gradually diminished with the aid of a small syringe. If, when the gorge circle is no more than 30 mm in diameter, we take care to remove the oil with only very small portions at a time, we succeed in reducing this diameter to 27 mm and obtain the result represented in Fig. 46.

~J \. :"/"-,." .'

.

..

.-

=' ... "".... -

-' ,' ._ . ...

I

·1"·

.....

,

.

. ...

'"'

~.

~.

,.

Figure 46 Thus, it is evident that all these waisted figures with convex bases, which can, as with those studied in the previous sections, differ as little from the cylinder as we please, are also portions of the unduloid, but taken, on the contrary, in the indeterminate unduloid: while the equator of the inflation is in the middle of the former, the middle of the others is occupied by the gorge circle of the waist; the largest of the former consist of the whole inflation between two half-waists (Figs. 21, 22) (in the original text-tr.) and that represented in Fig. 46 consists of a whole waist between two portions of the inflations. §56. Now, take our horizontal rings again, and suppose that we can place the upper 1 This operation is performed by the method indicated in §46, i.e., by attaching the mass first to one of the fixed rings and then drawing it towards the other by means of a movable ring of the same diameter, and, finally, absorbing the superfluous oil.

SURVEY OF MINIMAL SURFACE THEORY

85

one closer or farther away from the other at will; in addition, form a cylinder between them, and then, without changing the distance between the rings, gradually remove the whole mass of the oil. If the ratio of the distance between the rings to their diameters is much less than that in the last experiment of the previous section, then the curvature of the bases, instead of augmenting with increase of the waist, on the contrary, decreases, 2

and if this ratio does not exceed approximately 3' then it becomes necessary to render the bases absolutely plane. If the ratio is still small, it is possible to go even further: continuing to absorb the liquid, we see the bases become concave; thus, for example, they form between our rings of 70 mm in diameter as cylinder of height 35 mm (Fig. 47); gradually absorbing the oil, we shall see the bases deform as the waist shrinks and, finally, lose all their curvature. We shall thus obtain the result represented in Fig. 48. If we continue to employ the small syringe, then the bases will acquire a concave curvature; but for the moment, let us dwell on the case when they are plane.

"""""".'1" ~ : ..

.:: ':....'" ; ...• ··.F

.....:. - ."

.. ~

Figure 47

J<. ·X Figure 48 With such bases, the waist contained between the rings can no longer (§49) belong to the unduloid, and we eventually obtain a new figure of revolution. Let us the refore investigate what this new figure is in its complete state. The bases of our partial figure being plane and, therefore, the mean curvature zero (§2), equilibrium requires that the surface of the portion contained between the two rings and, consequently, that of the remaining part of the complete figure, should also be of zero mean curvature; it is therefore necessary that

J... + J... =

0 everywhere, whence M N M = - N; and, as the quantities M and N can both be considered to belong to the meridional line (§36), this line must be such that, at each of its points, its radius of curvature should be equal in magnitude and opposite in direction to the normal. Geometers have shown that the unique curve which enjoys this property is the catenary line. 1 The curve then turns its vertex towards the axis to which the normals are related, the straight line which divides it symmetrically into two equal parts is perpendicular to 1 The catenary line is, as generally known, the curve that forms, in the state of equilibrium, a heavy chain which is perfectly flexible when suspended at two fixed points.

86

THE PLATEAU PROBLEM: PART ONE

this axis, and the vertex of the curve is at the same distance from this as the radius of curvature from the same vertex. Our figure is consequently in its complete state and could be generated by the revolution of a catenary line thus placed with respect to the axis. Because of this, we give it the name of Catenoid; Fig. 49 represents a meridional section which is large enough and whose axis of revolution is ZZ'. The catenary line being a curve with infinite branches, the catenoid therefore also spreads to infinity in similar fashion to the cylinder and the unduloid, but no longer only in the direction of the axis.

Figure 49 It is hardly necessary to make a remark that if, in the partial catenoid realized by us, the curved surface contained between the rings exerts the same pressure as the plane bases, this is because, in this curved surface, the mean curvature (§2), that is, the mean of all the positive and negative curvatures around the same point, being zero, its influence on the pressure is also zero, so that this pressure remains the same as if there were no curvature at all. We have already seen an analogous case at the end of §31. §57. By virtue of the principle which terminates §2, it is possible to conceive two catenoids of different aspects: one implied by Fig 49, in which the liquid fills the space left by the catenary line between itself and the axis of rotation and another, in which the liquid occupies the space enclosed by the curve. Fig. 50 represents a meridional section of the latter.

z

z· Figure 50

SURVEY OF MINIMAL SURFACE THEORY

87

§58. In the experiment of §56, we arrive, as we have already said, at the necessity of making the bases of the figure plane, only if the distance between the

f

of their diameter. We shall return (§62) to this rings does not exceed about experiment; but we obtain an important consequence which we now derive: it must be concluded, in fact, that, for rings of a given diameter, there is a maximum for the distance between the rings beyond which no portion of the catenoid is possible between them. We are going to show that this result is in accordance with the theory, and we shall be led, at the same time, to a new result. We have seen that the generating catenary line must satisfy the condition that the radius of curvature at its vertex should be equal in magnitude and opposite to the quantity which measures the distance from this vertex to the axis of revolution, this being, as we conceive it in the meridional plane, the straight line perpendicular to the axis of revolution and representing the axis of symmetry of the catenary lines, then another straight line parallel to the axis of revolution, and placed from it at a distance equal to the radius of the rings. For whatever position of the latter, their centres will remain on the axis of revolution, and their contours will always stand upon the second above-mentioned straight line which we shall call, for brevity, the straight line of the rings. Finally, imagine in the same plane a generating catenary line having its vertex at the point where the straight line of the rings is cut by the axis of symmetry which we have referred to. This catenary line will be tangent at this point to the straight line in question, and will consequently only be able to stand upon the rings when the contours of these pass through the point of tangency or, in other words when the distance between the rings is zero l ; the catenary line under consideration corresponds, therefore, to the case when the rings are not moved from each other at all. Now, suppose that the curve leaves this position, and gradually moves towards the axis of revolution, changing so as to satisfy, at each moment, the condition of the equality between the curvature radius at its vertex and the distance from the vertex to the axis; in each of these new positions, it will cut the straight line of the rings at two points which we denote by A and B. The distance between them will represent, therefore, in each of these very positions, the distance between the rings and the corresponding catenary line will represent the meridional line of a catenoid whose portion is contained between the rings. With this in mind, let us examine how the points A and B move. In the initial position of the catenary line, that is, when its vertex is tangent to the straight line of the rings, the points A and B coincide with the point of tangency; then, at the moment the vertex of the curve starts moving towards the axis of revolution, these two points separate, and move away from each other. So, I claim that the distance between them will attain a maximum after which it will, on the contrary, start decreasing. In fact, by the condition, to which the catenary line is subjected, when its vertex almost touches the axis of revolution, the radius of curvature at this vertex will become very small, whence it follows that the two branches of the curve will be very near to each other and that, consequently, the two points A and B will also be very near to each other; finally, when the vertex is on the axis, these same points will again become coincident, since then the radius of curvature at the vertex is zero and, thus, the two branches of the curve will form only one straight line coincident with the I

For simplicity, we shall regard the rings as formed of widthless wire.

88

THE PLATEAU PROBLEM: PART ONE

axis of symmetry. The points A and B, in moving from the coincidence position, first move away from each other, and then draw together to become coincident for the second time, whence, of necessity, the distance between these two points attains a maximum; moreover, it is easily seen from the nature of the curve that this maximum must be finite, and even must not be considerable relative to the diameter of the rings. It is evident that, in its path up to the axis of revolution, the curve has passed through all the positions which, with the rings given, can lead to equilibrium; the above maximum constitutes, therefore, a limit for the distance between the rings, beyond which no catenoid is possible between them. But what is happening provides us with another equally remarkable consequence. Since, in passing the path of the vertex of the catenary line, the points A and B first move away only to approach each other later, they again necessarily pass through the same distances so that, for each distance less than the limit, they belong to two catenary lines at once; so, it follows that to any distance between the rings less than maximum, there always correspond two distinct catenoids standing upon these rings, but penetrating them. It is seen without difficulty that the vertices of the two generating catenary lines, which for zero distance between the rings, are, one on the common contour of the rings in contact, and the other on the axis of revolution, draw together more and more as the rings move farther away from each other, and finally coincide, together with the two entire curves, when the distance between the rings attains its maximum. The two catenoids will therefore differ by a small amount as the distance between the rings becomes large, and will coincide in the limit. §59. All the catenary lines are, as is generally known, similar to each other; if, therefore, we consider a sequence of complete catenoids generated by the catenary lines of different dimensions, then all these catenary lines will also be placed similarly with respect to the axis of revolution in accordance with the condition which they have to satisfy (§56) and, consequently, all the catenoids will become similar figures. The complete catenoid is not, therefore, susceptible to variation of form as the unduloid, but constitutes a unique figure, like the sphere and the cylinder. Thus, the two complete catenoids which, theoretically, stand upon the same rings when the distance between them is within the limit, do not differ one from another except in their respective absolute dimensions. §60. Of the two partial catenoids belonging to these two complete catenoids and (equally possible according to the theory), between the rings, our procedure of necessity gives the least reentrant one; and if we then try to arrive at the most reentrant one by removing new quantities of oil in mass, it is always, as we shall soon see, another figure of equilibrium that is produced; we can therefore make the justifiable conclusion from the impossibility of realizing this most reentrant partial catenoid that it would constitute a figure of unstable equilibrium. As to the least reentrant, it evidently forms a portion of the complete catenoid that increases as the distance between the rings approaches the maximum. This is because, as the rings move farther away from each other, the arc of the catenary line that they intercept between them (§58) becomes a more considerable portion of the curve. To make a partial catenoid larger with respect to the complete catenoid, it would be required that the catenary line should penetrate deeper between the rings; but from that moment, for

SURVEY OF MINIMAL SURFACE THEORY

89

any small quantity that the vertex of the curve advances, the distance between the rings would diminish (ibid.) and there would be another, less reentrant catenary line possible standing upon the same rings, and the partial catenoid generated by the first being the more reentrant, it would be unstable. The catenoid of largest height constitutes, therefore, the largest portion of the complete catenoid which can be realized between two unequal rings. We indicate here another consequence, to which the above argument seems to lead, and which would contradict the facts: for any distance between the rings less than maximum, the least reentrant catenoid always seems to be perfectly stable, and, as we have seen above, the most reentrant one must be regarded as being always unstable; so, the catenoid of the largest height forms, as we have also seen, the passage between the catenoids of the first category to those of the second and, consequently, between stable and unstable catenoids; we can therefore state that the catenoid of largest height is at the limit of stability of this kind of figure; meanwhile, when we realize it with a mass of oil (§62), it manifests a very definite stability. We shall find out in the chapter where we deal with stability questions, what this apparent contradiction depends on. §61. It is easy to see that the third limit of the variations of the unduloid, about which we spoke in §51, is nothing but the catenoid. In fact, making the partial unduloid vary in the manner indicated in the same section, it is clear that as the volume of the mass increases, the relative normal and the relative radius of curvature at the vertex of the convex meridional arc grow, and become infinite simultaneously with this volume; whence it follows that, in this limit, the quantity

~+~ M

N

is zero, which we know

to be characteristic of the catenoid.

+ ~ thus converging to zero as the unduloid approaches the M N catenoid, the pressure exerted by the superficial layer converges at the same time to that of a plane surface; if, therefore, we consider between two rings a waist belonging to an unduloid, and if we imagine that this unduloid moves by degrees towards the catenoid, then the bases of the figure, whose pressure must be always equal to that of the waist, will of necessity become less and less convex and, finally, will be entirely plane. This then is what realizes, evidently, the experiment of §56: when, after having formed between the The quantity

~

2

two rings a cylinder whose height does not surpass 3" of the diameter, we gradually remove the liquid and, as the bases deform gradually until all their curvature has been lost, the waist that is formed and gradually becomes more considerable clearly belongs to an unduloid which tends to its third limit; and the experiment in question thus helps to pass progressively from the unduloid to the catenoid. If we combine what precedes with the contents of §55, we shall be able to derive this justified conclusion: that each waist standing upon the two rings and presenting the convex bases is a waist of an unduloid only if the curvature of the bases is greater than, equal to, or less than that of the bases of the cylinder contained between the same rings. §62. I have tried to determine experimentally, at least in an approximate manner, the maximum ratio between the height and the diameter of the bases. The exterior diameter

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THE PLATEAU PROBLEM: PART ONE

of the discs employed was 71 mm: In each of the attempts, a cylinder started being formed between these rings, then the oil was removed in the mass by syringes filled beforehand and afterwards in small portions; the operation was interrupted from time to time for observations. In this way, it was found that the largest distance between the rings, for which a figure with plane bases could be obtained, was of 47 mm, Le., sensibly

f of 71. We can conclude that the maximal height of the partial catenoid is either exactly or almost equal to f of the diameter of the bases. This catenoid is represented in near to

Fig. 51.

Figure 51 These experiments have presented curious particulars, a description of which will be given in Chapter IX. We terminate the study of the unduloid and the catenoid here, and we now pass on to yet another figure. §63. We have already caught sight of a portion of this figure: it is the waist with concave bases, about which we spoke in §56; this waist is foreign, by the nature of its bases, to the unduloid and the catenoid. To realize it, it is required, as we have said, that

f

the distance between the rings should be less than of the diameter; Fig. 52 represents a similar waist in the meridional section, with the distance between the rings equal to about one-third of the diameter, and after the bases have already sunk strongly; the dotted lines are the sections of the planes of the rings. Let us now try, as we have done with respect to the two previous figures, to determine the complete form of the meridional line.

· · · · · _ ·_····-·-l·a J--~ .

"

.

........--...::.. .. .

..:::............. -........ b Figure 52

We first make some remarks. In the first place, suppose that the ratio of the distance between the rings to their diameter is sufficiently less than to permit one extraction of a large quantity of liquid without the fear of occasional decomposition. In the case, the waist and the bases both deform more and more, and it becomes clear that there must be a moment, after which their surfaces would not be able to co-exist without intersecting each other. We shall see in Chapter VI what occurs then; but we see that if from this moment we want to observe the waist in all the phases when it belongs to the new figure of equilibrium, it is necessary to provide an obstacle to prevent the base from deforming,

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and this is what we achieve without difficulty by substituting discs for the rings; it is then possible to remove the oil until the figure decomposes spontaneously in the middle of its height. The discs which I have employed are, as for the rings, 70 mm in diameter; the lower one rests on three legs, more solid than those of the rings, and emanating from points situated between the boundary and the centre; the upper one is sustained by a sufficiently thick iron wire fixed normally to its centre. In the second place, the waist, which may be formed between the rings or shifted farther between the discs, always looks perfectly symmetrical on either side of its gorge circle. Moreover, it is what is required by the theory, since the mode of reasoning of §48 is independent of the nature of the meridional line and can be also applied to the waist, with which we are as much occupied as with a waist of the unduloid. If, therefore, in a meridional plane, we consider a straight line perpendicular to the axis of revolution, and passing through the centre of the circle of the gorge, then all that the complete meridional line will offer on one side of this straight line, it will also offer, in an exactly symmetrical manner, on the other side so that this very straight line will constitute an axis of symmetry. In the third place, since, in employing the rings, the bases of the partial figure are concave spherical caps and hence have negative mean curvature, the same must be true of the surface of the waist and, hence, of the rest of the complete figure; thus, . - 1 + -1.is everywh ere negative. . . t h·is comp Iete fi19ure, t h e quantity M N §64. With these preliminaries over, I claim that the points a and b (Fig. 52) where the partial meridional line stops cannot be points of inflection in the complete meridional line. We see, in fact, by the direction of the tangent at these points, that if the meridional line assumed from these points a curvature of contrary sense (Fig. 53), then the radius of curvature would be directed towards the interior of the liquid in this part of the figure In

(just like the normal), and that the quantity

~ M

+

~ would therefore become positive. N

z' Figure 53 Beyond the points a and b, the meridional line starts, therefore, to maintain a concave curvature; and the same sense of curvature is evidently maintained for the same reason, as the curve moves away from both the axis of revolution and the axis of symmetry. But the curve cannot continue moving away from these two axes indefinitely: in fact, if its

92

THE PLATEAU PROBLEM: PART ONE

movement were like this, then it is clear that the curvature would have to diminish so much as to vanish in each of the two branches at the point at infinity with the effect that, at this point, the radius of curvature would have an infinite value; and, as the same would evidently hold with the normal, the quality

~ + ~ would become zero in this limit.

M N It is, therefore, very necessary that, at a finite distance from its vertex, the curve have two points at which its elements should be parallel to the axis of symmetry. This, as we shall see, is also confirmed by experiment.

§65. Ifwe employ discs which we place at a distance equal to about one-third of their diameter, and if the absorption of the liquid has gone sufficiently far, then the angle made between the latter elements of the surface of the mass and the plane of each of the discs diminishes, and then completely vanishes, so that this surface is then tangent to the planes of the discs (Fig. 54), and thus the latter elements of the meridional line are parallel to the axis of symmetry. It is very difficult to judge the exact point where this result is attained; but we can make sure that it actually occurs while continuing to remove the liquid: we shall in fact shortly see the circumferences that which terminate the surface of the mass abandon the boundaries of the discs, withdraw, with the diameter decreasing, to a certain distance inside these boundaries, and leave a small zone of each of more solid ones free; so, as these zones remain thoroughly soaked with oil, though in an excessively thin layer, it is clear that the surface of the mass must abut them tangentially.

Figure 54 If the distance between the discs is still small, then we obtain a result of the same nature; we can make the circles in contact narrower or, in other words, increase the width of the free zones (Fig. 55) before the spontaneous decomposition in the middle of the figure occurs.

Figure 55

§ 179. This is, in fact, what I have verified by experiment, employing the method described above (§ 175), that is, by tracing on paper in large strokes (in the manner indicated) the base of a system of three hemispheres with their partitions, then placing above it a glass plate soaked with glycerine solution and, finally, setting on it, over the three partitions of circles, three bubbles, at first a little too small, and then successively increasing them by manoeuvring slightly in the above-mentioned manner, and simultaneously shifting the glass plate on the paper if necessary. In this way the base of the laminar system thus realized is completely covered by the drawing. The graphical constructions that I have listed above for the bases of systems of spherical caps implicitly assume the inverse proportionality of the pressure to the

SURVEY OF MINIMAL SURFACE THEORY

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diameter; the exact coincidence of the bases of the realized systems with the traced systems provided, therefore, a new verification of the validity of this law in addition to the direct one obtained in § 12l. These are the experiments, to which I alluded in the section just quoted. These same experiments confirm also, at least with respect to films of spherical curvature, the independence of the tension of the curvature (§ 159). §180. If we imagine that a fourth spherical cap is attached to the system of the previous three, then we shall be able to consider two different dispositions of the set, let alone, of course, the one in which the fourth cap is placed so as to be united only to one of the others. One of these dispositions contains four partitions joined by only one edge, and the other contains five joined by two edges. To simplify the question and the graphical constructions. I will suppose the four caps to be equal in diameter, in which case the partitions will evidently be plane. Then, it is possible to assume, in the first place, that the four caps are joined so that their centres are placed as the four vertices of a square, which will yield the system whose base is represented in Fig. 56, where there are four partitions abutting on the same edge at right angles; this system is evidently one of equilibrium, since everything is symmetrical.

Figure 56 We can assume, in the second place, that after three caps are united, the fourth one is united to two of them; in this arrangement, the four centres will be at the vertices of a rhombus, and we shall have the system whose base is represented in Fig. 57, where there are five partitions. This system is evidently a system of equilibrium too because of its symmetry; but here, on the same liquid edge, there abut no more than three partitions making the angles of 120°. So, if we try to realize the first of these two systems on the glass plate, then we shall not succeed or only obtain it at an inappropriate moment, and rapidly pass to the second one.

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THE PLATEAU PROBLEM: PART ONE

Figure 57 As to the last, we obtain it directly without any difficulty, and it is preserved. We must conclude that the equilibrium in the first system is unstable, and, thus, it becomes probable that the four partitions abutting on the same edge cannot co-exist. Moreover, we remark that in the film assemblage of Fig. 61 (in the original text-tT.), the liquid semi-circular edge which unites the two spherical caps is common only to three films, namely, these two caps and the partition and, as we know, these three films form equal angles between each other; even in the assemblage of Fig. 66 (in the oTiginal text-tT.), each of the liquid edges which unite the caps pairwise is common only to three films making equal angles between them (as does the edge joining the three partitions); finally, this is what also takes place, evidently, in the assemblage of Fig. 57. §181. We remark, on the other hand, that, in the systems of Fig. 66 (in the original texttT.) and Fig. 57, there are four edges converging at one point, namely, the three which unite the caps pairwise and the one which unites the three partitions; thus we easily deduce from the equality of the angles formed by the three films united by the same edge, that the four edges, which converge a tone point, also form equal angles at this point. In fact, if we suppose at first that the films are plane and, consequently, that the edges are rectilinear, then we shall evidently get that the system constitutes a set of four trihedral angles in each of which three dihedral angles are equal; so, by virtue of a well-known theorem, this equality entails, for anyone of these trihedral angles, that of the plane angles; but, since each of the latter is common to two of the trihedral angles, it follows that in the system all the plane angles, that is, those formed by the four edges, are equal. The films, or at least, part of them, are curved in our systems of caps and partitions and, consequently, the same is true for the edges; but we can evidently replace the curved films by their tangent planes at the point common to the edges, and the curved edges by their tangents in which these planes intersect. Conversely, if the four edges make between them equal angles at their common point, then it follows from the symmetry of the set that the films must also make equal angles with their common edges. Thus, in all cases, the equality of angles formed by the four

SURVEY OF MINIMAL SURFACE THEORY

95

edges converging at a common point, and the equality of the angles formed by three films with the same edge follow of necessity from each other, at least in the immediate vicinity of the point in question. Let us now find the common value of the angles between the edges. In accordance with what we have said, it suffices to consider four straight lines converging at the same point at equal angles. Then, to follow the simplest way, we consider a regular tetrahedron abcd (Fig. 58). Let 0 be the centre of this tetrahedron. Draw the straight lines oa, ob, oc, od to the four vertices; these straight lines will evidently make equal angles between them, the value of which will consequently be the one required. Then produce the straight line oa to meet the base at p and join pd; the triangle opd will be right-angled at p. Now, we observe that the point p is the centre of gravity of the tetrahedron and that the point p is the centre of gravity of the base bcd; it is known that if, in any pyramid, the vertex is joined to the centre of gravity of the base, then the centre of gravity of the pyramid is situated on this straight line at three-fourths of the length from the vertex; therefore, op is one-third of oa, and, since the point 0 is equidistant from the four vertices, op is also one1 third of od. Consequently, in the right triangle opd we have cos dop = 3' whence, finally, cos doa =

3

Thus, when four straight lines converge at the same point at equal angles, each of these

+

angles is the one having for its cosine, from which we find that this angle is 109 0 28 / 16", that is, very near to 109 degrees and a half.

Figure 58 Hence, this is the value of the angles at which the liquid edges converge at the same point in the film assemblages that we have studied so far. §182. These systems resulted simply from attaching spherical caps; but we can produce them in a much more curious way: we saw in Chapters III and IV the ease with which films of glycerine solution are developed with the support of iron wire contours; hence, if we take an arbitrary wire polyhedron of those used in the realization of oil

96

THE PLATEAU PROBLEM: PART ONE

polyhedra (§29), and which is slightly oxidised, plunge it into the glycerine solution, and, after leaving it there for several seconds for better soaking, remove it, then we must find, and we will, in fact, find it occupied by a system of films; this system which we would expect to see having an irregular structure as a consequence of having moved the skeleton figure haphazardly while taking it out of the liquid, will, on the contrary always have a perfectly regular and symmetric disposition. This disposition varies with the form of the wire model; but, in general, it is invariably the same for the same wire polyhedron. For example, the wire cube always yields the system which I represent here (Fig. 59): it is composed, as we see, of twelve films emanating from twelve solid edges and will converging at a unique small quadrangular film placed in the middle of the arrangement. The sides of this small film are slightly curved, so that all the other liquid edges, and, therefore, all the films except the central one, have small curvatures. The curvatures of liquid edges which emanate from the vertices of the wire cube are too little pronounced to indicate them in the figure. In performing the experiment one must take care not to plunge the whole of the fork supporting the wire model into the liquid, since otherwise a film is then also formed on this fork and may modify the system of the other films. Recall that my wire cube has a side of 7 cm.

Figure 59 §183. The laminar systems thus developed in wire models have excited admiration in all the persons to whom I have shown them: they are, as I have said, of a perfect regularity, their liquid edges are extremely fine, and their films display extremely rich colours after some time. To observe one of them in a convenient manner and without agitating it, we place the wire model with this film on a small frame of iron wire similar to that represented in

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97

perspective in Fig. 60; the frames that I used were four cm wide, twelve long, and two and a half high. After each series of experiments, the wire models are washed with rain water and allowed to dry on filter-paper. We have to mention here that these laminar systems can be realized, as a last resort, by replacing the glycerine solution by a simple soap solution. The figures are then not very durable, but we can always dip the skeleton figure into the liquid again as often as we please, and renew the observations.

Figure 60 §184. Now, whatever polyhedron whose edges form the skeleton is employed, if we attentively examine the laminar system produced, we shall discover that three films are always adjacent to the same liquid edge and that the liquid edges converging at one "liquid" point are always four in number; thus, the numbers which we have already found in the systems of caps are also found in all the laminar systems, so that their invariability leads to two laws. Moreover, here, just as for caps, it follows from the equality of the tensions that the three films united by one liquid edge must necessarily form equal angles with this edge and, hence, the four edges concurrent at the same "liquid" point also make between themselves equal angles, which is what constitutes two other general laws. § 185. Among the wire models which I submitted to the experiment, three provide systems composed of plane films, namely that of the regular tetrahedron, equilateral right triangular prism and the regular octahedron; these permit us to establish the equality of the angles formed by the liquid edges, since they can be measured; consequently, the equality of the angles between the films can be also verified. I reproduce here (Figs. 61, 62, and 63) the diagrams of these three systems. To understand these diagrams and most of those to follow, we must not forget that each of the solid edges emanates from a film and is directed towards the interior of the wire model, being attached to the other films by the liquid edges which are represented in thin lines in the figures. The system of the tetrahedron is, as we see, formed simply by six films united along four liquid edges all meeting in the centre of the figure; that of the triangular prism consists of two pyramids standing upon the bases of the skeleton figure and whose vertices m and n are at the extremities of one liquid edge parallel to the sides of the prism; finally, that of the octahedron, one of the most beautiful that can be obtained, contains, in addition to the twelve films emanating from solid edges, six quadrilaterals elongated in the direction of one of their angles, each having the vertex of this angle at

98

THE PLATEAU PROBLEM: PART ONE

one of those of the wire model, and the opposite vertex at the centre of the figure; these quadrilaterals are disposed so that, in each pair meeting at two opposite vertices of the wire model, their planes are at right angles to each other. It should be borne in mind that, in the wire octahedron, other systems can form; we shall speak of them below. In the system of the tetrahedron (Fig. 61), the equality of the angles is obvious, everything being perfectly symmetrical in all the directions; besides, we see that the four

Figure 61

, Figure 62

SURVEY OF MINIMAL SURFACE THEORY

99

Figure 63 liquid edges occupy precisely the positions of the straight lines oa, ob, oc, od of Fig. 58. In that of the triangular prism (Fig. 62), the angle between the vertical edge mn and the

+

oblique edge np, for instance, must have for its cosine, from which it follows that the vertical height of the point n over the plane of the base is one-third ofnp; from which we easily deduce that, designating the edge pq of the base by a, this vertical height equals

a

2../6'

Therefore, if b denotes the lateral edge of the skeleton figure, we

have:

mn = b -

a

../6'

In the wire model l that I have employed, the edge b was 70.70 mm, the edge a 69.23 mm; substituting these numbers into the above formula, we find mn = 42.44 mm; the reading of the cathetometer was 42.37 mm, the value which differs from the previous only by 0.07 mm, i.e., by less than two thousandths of one or the other.

1 It is necessary to explain here how these evaluations were obtained. A model of iron wire cannot be geometrically perfect, and we have separately measured each of the lateral edges; after taking the mean of the results, we have done the same for the base edges; but these measurements have not been performed on the edges directly, because the terminal points of these were more or less masked by the soldering; we have proceeded thus: with the wire model placed vertically, and one of its lateral faces opposite the cathetometer, we have determined, by sighting at a straight line near the vertical edge as accurately as the soldering seams permitted, the distance between the two horizontal edges. We have then done the same thing near the left vertical edge; having turned the prism about its axis so that the other two lateral faces were successively presented to the cathetometer, we have repeated the same operations on each of them; thus, we have had six slightly different values for the length of lateral edges, their mean having found to be 69.83 mm. This done, we have placed the wire model horizontally on convenient supports so that one ". , - " . , ' 'orpo .,,"<
100

THE PLATEAU PROBLEM: PART ONE

As to the system of the octahedron (Fig. 63), it presents, in each of the quadrilaterals, of which I have spoken, a dimension to be measured easily by the cathetometer: it is the distance between the vertices t and u of two obtuse opposite angles. On the other hand, proceeding from the principle that these two angles, as well as that whose vertex is at the centre of the figure, must have

-

2.

for their cosine, we easily arrive at the

3

demonstration that this distance tu must be exactly one-third of the length of the edges of the octahedron. To see this, let stou (Fig. 64) be one of the quadrilaterals; draw the two diagonals tu and os intersecting at right angles at m; denote the distance tu by d, the common value of the three obtuse angles by ct, and the length of the edge of the octahedron by a. In the triangle ots, the angle at t being ct, and the angle at the angle at s will be 180 0

-

i

2

0

being .!..ct, 2

ct; hence, the two right triangles omt and smt yield

tc..--=+--~

Figure 64

bounded the model from the right and from the left were vertical, and we have proceeded with respect to these in the same manner as with the previous ones; then, turning the prism about its axis, we have performed the same operations with the other two edges, so that we have also had, for the edges of the bases, six values whose mean is 68.36 mm. But these two means had to undergo a small correction; in fact, the liquid laminas do not touch the solid wires along their generators the lengths of which we have measured, but along other generators situated farther inside the wire model; from which it follows that the above two numbers are very slightly less than in reality. To perform the correction in a simple manner, note that, by virture of the tension, the plane of each of our films should pass through the axis of the solid cylindrical wire which supports the film and that, consequently, the liquid edges produced will meet at the points where these axes intersect; we can, therefore, without changing the laminate system in any manner, and, therefore, the length of the liquid edge mn, mentally replace the set of axes of the solid wires, constituting our wire model, by wires; which evidently leads to adding the diameter of these wires to each of our two numbers; so, this diameter, also determined by means of the cathetometer, has been found as the mean of six measurements taken on the different wires, and equals 0.87 mm; it is by the addition of this quantity to our two numbers that the values ofb and a given in the text have eventually been found.

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SURVEY OF MINIMAL SURFACE THEORY

.!.d om

.!.d

2 1 tan 2

and IX

2

ms =

tan (180 0

3 2

-

-

Now,

note

that

the

diagonal

IX)

os, or the sum of the two lengths om and ms is half the height of the octahedron, and consequently, as it is easy to see, equal to ~ we have: 1 ( - -1- + -d 2 1 tan - IX tan (180 0 2

But, knowing that the cosine of the angle and tan (180 0

-

i2

1

IX

1

is - - we find that tan -IX = 2 3' 2'

IX)

=

.!. .J2.

5 These values are substituted in the above formula, and we obtain, as predicted, that

d=~ 3

In the wire model I considered, measuring gavel the value 69.49 mm for a one-third of which is 23.16 mm, and 23.14 for the distance d. The difference 0.02 mm between the value calculated and the value measured is, as we see, even less significant than in the previous case. § 186. The laminar systems of other skeleton figures, i.e., those which contain curved films and, consequently, curved liquid edges, also prove the equality of the angles at which these edges meet at the same "liquid" point, in a manner quite curious, though a little less precise; besides, they prove the equality of the angles between the three films which join along each of these same edges, at least in the immediate vicinity of the point under consideration (§ 181 ). As the first example, take the laminar system of the wire cube represented in Fig. 59. Each of the angles of the small central quadrangular film must be of 109 and a half degrees, i.e., greater than a right angle from which it follows that the sides of this small

1 A regular octahedral wire model can be considered as being formed by the set of three squares made of iron wire and whose planes intersect along the diagonals; therefore, the skeleton figure was placed so that these squares were presented, one after another, opposite the cathetometer with two of their sides directed virtically and, in the three positions of the skeleton figure, we measured, as near to each of these vertical wires as possible, the distance between the two horizontal wires. We obtained twelve values whose mean was 68.59 mm; moreover, we took eight measurements of their diameters on the different wires the mean of which was 0.90 mm. In accordance with the considerations set forth in the previus note, we added this diameter to the above number, and thus found the value of a given in the text. As to the value of d, since the small irregularities of the wire model had to introduce slight differences between the six quadrilaterals, we measured this distance in each of them, and the value of d given in the text is the mean of these six measurements.

102

THE PLATEAU PROBLEM: PART ONE

film cannot be rectilinear and must have the form of arcs slightly convex towards the exterior; this is, in fact, what the realized system shows. I have represented in Fig. 74 the laminar system of the equilateral triangular prism, and I have said that, designating the length of the edges of the bases by a, the height of each of the film pyramids that stand upon these bases is equal to 2~ 6; but it is clear that, if the height of the prism is less than twice this quantity, or, in other terms, less than 0.4 of the base edges, then the system in question could not be produced. In this case, the analogue with other systems of which I shall speak below has led me to the prediction that the system will be composed of one equilateral triangular film parallel to the bases, placed at equal distances from the latter and attached to all the solid edges by other films; but as the angles of this film triangle must be of 109 and a half degrees, and the angles of an equilateral triangle with rectilinear sides are only of 60°, the sides of our triangle of films should of necessity convex towards the exterior, as for those of the small film of the system of the cube, but their curvatures should be much more pronounced; this is what the experiment has revealed fully, with the difference that the curvatures were seen to be considerable only in the vicinity of the vertices. The height of my wire model was about one-third of the length of the base sides; Fig. 65 represents the system seen from below; this system gives, therefore, a second example in support of the proposition stated.

Figure 65 In the third place, if we take as a wire model a regular pentagonal prism whose height is not too large relative to the dimensions of the bases, then the laminar system presents a pentagonal film parallel to the bases in the middle of its height, and to which, as with the triangular film of the previous system, all the films emanating from the solid edges are attached; the angle of two sides adjacent to a regular pentagon are of 108°, that is, very near to our angle of 109! 0; it follows that the sides of the pentagonal film must be reasonably straight, and that is what the experiment confirms: in the wire model that I have used, the edges of the bases are 5 cm long, and the height of the prism 6 cm; the sides of the pentagonal film are about 2 cm; and the eye cannot discern any curvature there. In this system, the films that emanate from the solid edges of the bases and are attached to the sides of the central pentagon seem to be plane, and this is what must occur, because they are supported on one side by the rectilinear edges of the bases and, on the other, by the reasonably straight sides of the central film; it follows from this that the oblique liquid edges which go from the vertices of the bases to those of the central pentagon seem to be straight. As to the triangular films which emanate from solid vertical

SURVEY OF MINIMAL SURFACE THEORY

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Figure 66 edges, they must be strictly plane because of the symmetry of their position. The system of which we speak is represented in Fig. 66. Finally, in the fourth place, if the wire model is ofa regular hexagonal prism, then the laminar system is analogous to the previous one in its general disposition, the central film certainly being hexagonal; but, as the angle between two adjacent sides of a regular hexagon is 120 0 , that is, noticeably greater than our angle of 109! 0, the sides of the film in question must be reasonably curved towards the interior, and this is what is also demonstrated by the realized system. The height of the model that I employed is equal to seven sixths of the distance between two opposite sides of the base or, in other words, the diameter of the circle inscribed in this circle. § 186b. In addition, we can form a system with curved liquid edges, in which the angle between these edges can be measured directly and precisely: in a large ring of iron wire supplied with three legs, we realize a plane film of glycerine solution, then we lay two small bubbles of the same liquid, reasonably equal in diameter, on this film near to one other. Each of them tends to be transformed in a bi-convex film lens encased with its boundary in the plane film; but, because of their contact, these two lenses are interlaced with a partition. With the exception of a special case they will not be of the same diameter but, by evacuating a small portion of air contained in the greater of them by means of a sharpened tube whose point has first been soaked in glycerine solution (see the note to § 175), we can eliminate the difference; then the partition is plane, and the system seen from below presents the aspect of Fig. 67; ab is the projection of the partition. The ring which I have used has a diameter of 20 cm; it is formed from an iron wire 3 mm thick and the legs are 8 cm high. To effect the stated verification experimentally, we have traced the projection of the system on a sheet of paper proceeding from the theoretical value of the angle at a; the procedure has been very simple: the angle between

104

THE PLATEAU PROBLEM: PART ONE

Figure 67

two tangents at a must have, as we know, -

~ for its cosine, and that between the two 3

radii joining the point a to the centre of the two lenses is its supplement; we have constructed the latter angle and marked off a length of 4 cm from a on each of its sides; then, taking the two points thus determined as centres, we have traced two portions of circumferences terminating at the points a and b; finally, we have drawn the straight line abo We imagine that if, after having realized in the ring a system a little smaller than this drawing, we place the same drawing between the legs of the ring, and, closing one eye, hold the other at a convenient height over the set, then we shall be able to see if it is possible to shift the assemblage of the lenses so as to project them exactly on the drawing traced. Meanwhile, the experiment performed in this manner will present difficulties, because the assemblage of the lenses, by following the small agitations of the surrounding air, changes its position incessantly. To eliminate this inconvenience, we have preliminarily held a thin wire across the ring along a diameter, so that it had to engage in the realization of the plane film, and we have laid the two bubbles in such a manner that the wire was also engaged in the partition; the system then lost its mobility. To proceed to the experiment, we have covered the drawing by a thin glass plate with the purpose of holding it; then, after having developed the plant film, we have placed the ring with its legs on this plate, and we have done this in such a manner that if the eye is placed well over the arrangement, then the wire is projected on the straight line ab of the drawing (we have produced the straight line on both sides); then we laid the two bubbles and equalized the two lenses with great care. Then, placing the eye conveniently, we could effectively observe the contour of the assemblage of the real lenses project exactly on the drawn contour. § 187. The fact, about which I have spoken (§ 180) demonstrating that a system in which more than three films meet at equal angles at the same liquid edge is in a state of unstable equilibrium, is related to a system which possesses simultaneously more than four edges meeting at the same "liquid" point. The instability could, therefore, be equally attributed to this latter circumstance, and it is required to decide whether it is

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exclusively explained by one cause or other, or by their combination only. For that, let us return to the experiment reported in §139, that is, we take the set oftwo rectangles that intersect at right angles in the middle of two of their opposite sides as a solid system; but, to render the effect more apparent and the wire model more convenient to handle, we give the two rectangles a smaller width and greater height (Fig. 68).

Figure 68 The simplest laminar system which can be imagined in this wire model is composed of four plane films occupying the four halves of the rectangles and meeting at the unique rectilinear edge ab (Fig. 69) that joins the two points of intersection of the same rectangles. This system, because of its symmetry, will evidently be a system of equilibrium, and will not have any "liquid" point common to several edges, but the edge ab will be common to four films. Thus, when we remove this skeleton from the glycerine solution, we never find it occupied by the system that I have indicated: in the one realized, in place of the edge ab, there is always, as with the wire model of § 139, a plane film (Fig. 70) terminating with two curved edges to which the curved films emanating from the solid edges are attached. Here, each of the two liquid edges is seen to be common only to three films and we have to conclude that the instability is, in fact; a property of laminar systems for which this condition will not be fulfilled. We add that Brewster who took pleasure in reproducing and varying my laminar systems has considered 1 a very curious modification of the above experiment: he has rendered the two rectangles mobile about their points of intersection, so as to vary the dihedral angles, and then, on gradually increasing those of the angles, of which the plane of the oval film is the bisector, he has observed this film shrink by degrees, so that, when 1 On the/igures 0/ equilibrium in liquid/ilms (Transactions of the Royal Society of Edinburgh, voI.XXIV, 1867).

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o

Figure 69

Figure 70 the angles in question have become equal to 135 0 , the oval film has reduced to a simple straight liquid edge joining the two points of intersection; but, at the same moment, the system is modified, and a new oval film is produced, which is the bisector of the other two dihedral angles. Brewster has therefore succeeded in realizing, but for an inappreciable instant, a system, in which four films are united by the same liquid edge, and this beautiful experiment completes the verification of the instability of a similar system.

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As to the second circumstance, I shall first make the remark that if, in the wire cube, we consider a system formed by twelve triangular plane films emanating, respectively, from twelve solid edges and meeting at the centre of the cube (Fig. 71); this system will necessarily be a system of equilibrium due to its perfect summetry, and we see without difficulty that only three films abut on each liquid edge, the films making equal angles between them; but there will be eight liquid edges meeting at the central point. Thus, we know that this system cannot be produced with the glycerine solution and we always obtain the one represented in Fig. 59 in which only four liquid edges meet at each of the vertices of the central quadrangular small film. We can conclude from this fact that instability is also intrinsic in each system in which the same "liquid" point is common to . more than four edges.

Figure 71 We can obtain, in a permanent manner, the system in Fig. 71; but, as in the instability case of § 138, by introducing a solid part in this system; it suffices, in fact, to stretch a very thin iron wire from a vertex of the wire model to the opposite vertex. Meanwhile, when we remove the wire model thus disposed from the glycerine solution, the system that occupies it is not immediately the one in question; in addition, it contains a small quadrangular film; however, it is much smaller than that in Fig. 59, and is by no means placed symmetrically with respect to the wire model, it being supported by one of its vertices in the middle of the solid diagonal; but we soon see it spontaneously diminish in size and completely vanish, so that the system then becomes as in Fig. 7l. The process finish here, and the system remains perfectly stable in this state when the small film is decreased sufficiently slowly; but often this decrease is faster, and then another singular phenomenon occurs: at the moment when the small film vanishes, we see another much smaller one form; it is situated on the opposite side of the solid diagonal and has its plane

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perpendicular to that of the first, being supported not by a vertex any more, but by the midpoint of one of its sides at the midpoint of the solid diagonal! then this second small film decreases and vanishes in similar fashion to the previous; in this case, therefore, the system attains its definite form only by a sort of oscillation. § 188. It is required, I think, to consider now how well the laws that I have discussed, are established for all the assemblages of liquid films. These laws lead to a very remarkable consequence: the foam which forms on certain liquids, for example, on champagne, beer, or agitated soap water, is evidently an assemblage of liquid films, composed of a great number of small films or partitions intersecting each other and enclosing small portions of gas; consequently, though everything seems to be left to chance, the foam must obey the same laws; thus, its innumerable partitions necessarily join each other three by three at equal angles, and all its edges are distributed so that there are always four meeting at the same point, making equal angles between them. I have verified these facts by the following experiment: the bowl of a pipe was placed at the bottom of a vase containing some glycerine solution and, holding the pipe in a slightly oblique manner, its tube was continuously blown through so as to produce a large number of air-bubbles of sufficiently large size rising in the liquid. The formation of a partitioned structure being built down at the walls of the vase and evidently of the same constitution as the foam, but whose different parts are of much greater dimensions, was determined thus, in the way children do with soap water; thus deep in the foam so that the eye may still orient itself, it can be asserted that everywhere the same edge was common only to three partitions, and that only four edges were ever adjacent to the same point. As to the equality of angles between these edges, there were certain places where three of these edges meeting at a point seemed to be almost in the same plane; but, studying the environment with attention, we noted that these edges bended strongly approaching their concurrence point. The generation of a similar structure is explained easily, and the same holds for the foam: the first bubbles of gas arrive at the surface of the liquid giving birth to spherical caps attached similarly to those with which we were occupied previously, and soon the whole surface of the liquid is covered by them; then the films which produce the subsequent bubbles of gas will of necessity lift this first assemblage, determining the formation of lower partitions, so that there are, after a short time, two systems of films stacked on each other; with the gaseous bubbles always coming, this set is, in turn, lifted, and so on, everything disposing with more or less symmetry in accordance with the differences in the volume of the successive gaseous bubbles and the distribution of the points where they reach the surface of the liquid, and the light structure consisting of partitions enclosing the whole volume of gas which constitutes the bubbles in the spaces separating the partitions acquires more and more height. If the bubbles are extemely tiny, the partitioned structure will consist of too small parts for the eye to distinguish them in general, and we will thus have a foam. §189. We now return to the laminar systems of wire models and complete their study. 1 My wire model has been accidentally deformed and then repaired, and the second small film produced was not quite placed in the above-mentioned manner; the difference arose, undoubtedly, from a small irregularity of the model, either before or after the repair.

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First we examine the manner in which the films that must constitute one of these systems are disposed when the model is removed from the liquid, and immediately after it has been removed. We began with the case of a wire prism removed in such a fashion that all its bases are horizontal. When the upper base is removed from the liquid, each of the solid edges of which it is composed will necessarily be followed by a film. Then, if the angle contained between two lateral faces adjacent to the prism is equal to or greater than 120°, that is, one of the equal angles made by three films meeting at one liquid edge, then the laminas emanating, as I have said, from all the edges of the base, will, as we shall see, have to remain attached to the vertical solid edges, and in such a way that the lower base will not leave the liquid. As an example, take the wire regular hexagonal prism, in which the angle between two adjacent lateral faces is 120°, and consider it when it is yet only partly within the liquid. Suppose, for the moment, that the films which emanate from the edges of the upper base return to the interior of the skeleton, in which case other films will of necessity emanate from vertical edges to meet the liquid edges which will unite the first ones. All these films will be attached to the liquid in the vase in small masses with transversal concave curvatures, lifted along their lower boundaries; it is clear that, by virtue of their tension, these very films will abut on the crests of these small masses along the vertical directions; the films which emanate from the edges of the base will consequently have to bend in descending towards the liquid, and they will therefore become convex towards the interior of the figure in the direction of their height. But, as their two faces will be in contact with the free atmosphere, it is required that their mean curvature should be zero (§97), or, in other words, at each of their points, the curvatures should be equal in magnitude and opposite in two rectangular directions (§2); therefore, since the films in question are convex in the direction of their height, they will be concave in the direction of their width; thus, because of their concavity and reentrant direction towards the interior of the wire model, our films will necessarily make, pairwise, equal angles greater than those of the faces of the prism and, consequently, greater than 120°, which, as we know, is impossible; thus, these same films will have to cling, as I have predicted, to the lateral solid edges, so that the whole of the model is not outside of the liquid. This deduction is fully confirmed by the experiment: when the wire regular hexagonal prism is removed from the glycerine solution in the indicated position, we simply obtain plane films occupying all the lateral faces until the lower base is removed. The thickness of the solid wires, which is close to one mm in my wire models, would seem, in reality, to be sufficient for establishing the independence of these films; but here it has nothing to do with the phenomenon: I have constructed a wire hexagonal prism in which the lateral edges were simple hairs, and things proceeded in absolutely the same manner. In this model, the disposition of which would be difficult to represent in miniature in a figure, the upper base is lifted by springs, so that the hairs which elongate in the liquid are always stretched. § 190. But it is clear that this cannot happen in the case of a wire model whose bases have at least six sides, since then the lateral faces make between them angles less than 120° and the films that emanate from two adjacent edges of the upper base and which will communicate along the corresponding vertical solid edge with the intermediate part of the liquid that soaks this edge, must stretch to detach themselves from this very edge

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and both be directed towards the interior of the wire model, so that the angle of 120° is formed between them; then also, they will develop, certainly, a third film emanating from the vertical solid edge in question, so that these three edges are united by one oblique liquid edge emanating from the point where three solid edges join. This is what is verified by the experiment as well: when a wire prism with a square base is removed from the liquid, we see the film take a reentrant direction from the moment when the upper base appears, and the effect is still more pronounced with the wire model triangular prism; we state, at the same time, that these films behave as I predicted in the previous section, that is, they bend to meet the surface of the liquid along vertical directions and are concave in the direction of their width. As to the regular pentagonal prism in which the angle between two lateral adjacent faces is 108° and, consequently, somewhat less than 120°, we imagine that the tendency of the films to reenter must be feeble, and that thus the thickness of metal wires, of which the solid edges of my ordinary wire models were made, is sufficient, in this case, to establish the mutual independence of the films. Also, with the wire pentagon represented in Fig. 66, we obtain, in removing it, only plane films in the lateral faces; but a wire model, in which the lateral edges were of a very thin iron wire, and then the films were reentrant, with the difference that, as it had to be, they were much less reentrant than in the two previous models. This wire model with thin lateral edges is represented in Fig. 72; the bases are supported by means of two handles a and b, by which the model is held when dipped into the liquid.

b

Figure 72 § 191. Let us now see how these diverse systems are completed when we continue to lift the models. First, take the case of the hexagonal prism, and suppose that the wire model has dimensions whose ratio is of the value indicated at the end of § 186. When the lower base emerges a film is formed, stretching from this base to the surface of the liquid, and which shrinks from top to bottom. Ifwe lift the model more, then we shall soon reach a point where the equilibrium of the latter film is not possible, since it then spontaneously and swiftly contracts, closes while separating the liquid from the vase, and starts forming a plane film in the lower base of the prism. But this plane film forms right angles with those occupying the lateral faces and thus cannot persist, in accordance with

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what has been said in the previous section; matters have to be arranged so that it should form angles of 120 0 with the lateral films. This is what is effected in the simplest manner; the plane film in question climbs in the interior of the model, and diminishes its size, while drawing to iselfthe alteral films; each of these folds in two at the liquid edge which unites it with the first, whereas other films emanating from each of vertical solid wires also attach to the previous ones along the liquid edges that unite them pairwise, and equilibrium is established when the central film has reached one-half of the height of the film, because then everything is symmetrical; we thus have the system that I spoke of at the end of § 186. It is clear that, in this system, the oblique films directed towards the central film can pairwise make angles of 120 0 only on the condition that they are convex with respect to their width towards the interior of the figure, which, moreover, entails the reentrant curvature of the sides of the central film; but, because of the mean curvature necessarily being zero, this convexity requires that the films in question should be concave with respect to their height, so that all the laws are satisfied and, in the realized system, we can effectively observe these two opposite curvatures of the oblique films. As to the central film and films which emanate from lateral solid edges, they are necessarily plane, because their position is symmetrical with respect to the others. We remark that, in this same system, the oblique liquid edges which emanate from the vertices of the central film do not exactly pass from the vertices of the two bases, but from points situated at a small distance from the latter vertices on the lateral solid edges; the reason is seen easily: because of the rectilinear form of the sides of the bases, the transversal convexity of the oblique films will not exist near them; it is therefore required, in order that it may set, that these films should abandon the lateral solid edges to assume their reentrant directions only starting at a certain distance from the bases; it is, therefore, only at this distance that the oblique liquid edges can be created. § 192. Let us now pass to the case of the pentagonal prism with thin lateral edges, quadrangular prism or cube, and triangular prism; in all these cases, as we have seen, the films emanating from the edges of the upper base assume reentrant directions from the moment this base comes out of the liquid. When these films begin to appear, the small masses lifted by their lower boundaries will of necessity describe a polygon on the surface of the liquid, with the same number of sides as the solid base, that is, in accordance with the wire being a pentagon, quadrilateral, or triangle. However, as the films under consideration are vertical in their lower part and they join at angles of 120 0 , the sides of the above polygons must make angles of 120 0 between them, which evidently entails that they should be convex towards the exterior; these sides also must separate the horizontal curvature of the films, which we know to be concave towards the interior of the figure and hence convex towards the exterior. This convexity of the sides of our polygons will clearly have to be very slight for the pentagon, more pronounced for the quadrilateral, and still more for the triangle. All this is equally verified by the experiment. Having established these first facts, let us consider the development of the laminar system separately in each of the three wire models, as they are lifted more. With the wire pentagon, the curvilinear pentagon delimited on the surface of the liquid first shrinks a little until a sufficiently large part of the height of the skeleton is outside the liquid, then enlarges again and, when the lower base is at the level of the liquid, the reentrant films will have attached themselves to the sides of this base with their lower

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boundaries. Meanwhile, these films do not occupy the lateral faces of the prism: they are slightly convex towards the interior of the model with respect to their height, they are pairwise united by liquid edges, each of which constitutes an arc of feeble curvature, standing with its two extremities upon those of a lateral solid edge. Finally, a little after the appearance of the lower base, the concluding phenomena are the same as for the wire model of the previous section, that is, the film produced between the base under consideration and the liquid starts occupying the same base in plane form, then rapidly rises, while drawing the other films to itself, to yield, eventually, the system in Fig. 66. With the wire quadrangular prism, the curvilinear quadrilateral delimited on the liquid decreases and vanishes; it is then replaced by a small horizontal liquid edge whose extremities leave two descending liquid edges drawing apart from each other; these three edges bound a vertical plane film parallel to two of the faces of the prism, and attached by other films to the solid edges. Things do not change in character when we continue lifting: the plane film in question simply increases its height and becomes larger and larger in its lower part until the lower base of the prism starts appearing on the surface of the liquid; then the two descending liquid edges are supported by their extremities at the mid-points of two opposite sides of this base; after it completely leaves the liquid, the film that has occupied it rapidly transforms into four oblique films, two trapezia and two triangles, completing the system. In the particular case of the cube, we thus have the system represented in Fig. 59. If the height of the wire model is greater than the length of the sides of the bases then oblique films which are supported by these latter are identically the same as for the cube, and the films emanating from lateral edges, as well as the central plane film merely have greater height. Finally, with the wire triangular prism, the curvilinear triangle at the surface of the liquid decreases faster, and vanishes when the model has appeared with only a sufficiently small part, so that the triangular pyramid, which must stand upon the upper base in the final system, is found to be complete; then, continuing to lift the wire model, we see a vertical straight liquid edge stretch from the vertex of this pyramid to the surface of the liquid; this edge is common to three films emanating from lateral solid edges. Things remain the same on lifting more: the three above-mentioned films and the vertical liquid edge merely increase their height until the lower base appears; then the film, which goes to this base, instantaneously converts into the second triangular pyramid thus completing the system in Fig. 62; I suppose, certainly, that the model is of sufficient height not to give the system of Fig. 65. § 193. Now, take a wire model symmetrical about an axis passing through a vertex such as of a regular pyramid, regular octahedron, etc., and remove it by this vertex. It is evident that, in this case, films occupying the faces meeting at the vertex under consideration will not be formed, since the space that they leave between themselves and the liquid will be free of air; it is required, therefore, that the films respectively emanating from each of the solid edges should certainly be directed towards the interior of the skeleton figure. If there are only three solid edges meeting at the vertex in question and symmetrically placed, then, just as for the tetrahedron or the cube removed by a vertex, it is clear that the films emanating from these three solid edges will be united by a unique liquid edge descending vertically from the solid vertex to the surface of the liquid in the vase; and that is what takes place in reality. With the regular tetrahedron, things happen in the

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same way until after the appearance of the base, then the system finishes in the same manner as in the regular prism, and gives the result of Fig. 6l. If there are more than three solid edges meeting at the vertex by which the model is removed, then additional films will necessarily have to be formed due to the fact of instability, of which I have spoken in §§180 and 187. As an example, take the regular octahedron. It is clear that the films emanating from the four solid edges will unite not along a unique liquid edge, but along two edges emanating from the vertex and terminating an auxiliary vertical film, so that three films making equal angles between them meet at each of these latter edges, The auxiliary film is destined to form the upper quadrilateral in the complete laminar system (Fig. 63). Until the square that is the common base of the two pyramids constituting the octahedron leaves the liquid, the arrangement of the figure preserves the same disposition; then, as we continue to remove the model, we see modifications being produced (they will take rather a long time to describe) and, hence, the system tends towards the shape represented in Fig. 73, where the two faces abc and a' b' c' are each occupied by a plane film. This shape is completely attained at the moment when the lower vertex of the wire model leaves the liquid; but a change immediately occurs and the system takes the shape as in Fig. 63. Though this change is very rapid, we can meanwhile observe how it is produced with sufficient attention by recommencing the experiment many times: the two films which occupied (Fig. 73), as I have said, the faces abc and a' b' c' start towards the interior of the skeleton figure while turning about the solid edges ab and a' b' and, at the same time, a quadrilateral is developed beginning with the lower vertex. First, it is very small; it grows until its upper vertex reaches the centre of the model, which then constitutes the lower quadrilateral of the final system; at the same time also, the vertices f and g of the curvilinear quadrilateral sfgc rise through a certain height; this quadrilateral shortens, its edges become straight and, finally, the upper quadrilateral of the same final system is

s

I~-f--+--~~

c

Figure 73

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formed. Fig. 74 represents the phenomenon in process, and at the moment when the growing quadrilateral has acquired one-half of its final height. It is easily seen from this figure how the four other quadrilaterals of Fig. 63 are generated.

Figure 74 In order that these phenomena should occur almost certainly, the wire model should be removed vertically; moreover, it is required that this model should be constructed well, the iron wires that compose it have the least thickness possible and, especially, that they should be united to the vertices of the octahedron in quite a clean manner, at least on the side which faces the interior of the wire model. As I have already said, the same wire model can also yield systems quite different from that in Fig. 63. These systems are formed by curved films; they are developed especially when the model is slightly inclined on being removed from the liquid: one of the two, for example, which is formed in my experiments rather often, contains in the middle a hexagonal film placed parallel to the two faces of the octahedron and having slightly reentrant sides (§186); these sides are attached to the vertices of the above two faces by triangular films and to the edges of the same faces by trapezoidal films; moreover, the vertices of the film under consideration are attached by the triangular film to the other solid edges. The different examples that I have given in detail will suffice to explain how the laminar systems are generated, and to show that the theory can take into account all the particulars of this generation. § 194. If we take care that there should be no air-bubbles at all on the surface of the liquid of the vase before dipping a wire model, the laminar system obtained will not yield any space enclosed on all the sides by the films, and thus the films will be in contact with free air with their two faces. In fact, if, in removing the wire model, and before undergoing the rapid modification that gives it its final disposition the system contained a

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space enclosed on all the sides by the films, then this space would have to be formed and grow with lifting the model; but this is impossible, since the air, which would have to fill it, would have no exit to penetrate; for the same reason, the system could not yield a space partly enclosed by the films and partly by the surface of the liquid in this period of its generation; finally, when the rapid modification takes place, the film or the films which then rise in the system will not find any space of the second species to complete the closure of films, and the complete system will of necessity satisfy the indicated condition. We state here that another law, namely that all the films constitute surfaces of zero mean curvature in the laminar of any wire model; this is evident since, in accordance with the above, all of them have two faces in free air. § 195. Let us return to the systems of wire prisms. In addition the facts which I have set forth, these systems have presented to me other equally curious systems, which I am going to report on here. The system obtained with the wire pentagon as in Fig. 66 is, as we have seen (§186), composed of reasonably plane films; if we consider the oblique films which emanate from two homologous sides of the two bases to be united to one of the sides of the central pentagonal films and we notice that these two oblique films must form between them an angle of 120 0 , we shall evidently see that, for the bases of given dimensions, an increase in the height of the prism entails a decrease in the spread of the central pentagonal film, and that there is a limit of height, beyond which the existence of this film is impossible. Supposing that the oblique films are strictly plane, we find without difficulty that the limit under consideration corresponds to the case of the ratio between the height of the prism and the diameter of the circle inscribed in the base being equal to ../3, i.e., 1. 732; but, because of the small curvature of the above films, we thus have only an approximate value; below, we shall find the exact value (§204). It must be asked, naturally, what becomes of the laminar system when this limit is surpassed? To determine it, I have constructed a wire model, in which the height was about 2! times the diameter of the circle inscribed, and it has given me a singular result: when it is removed from the glycerine solution as the lateral edges are made of ordinary wire, all the lateral faces are first occupied by plane films and, after it is completely removed, a plane film is also formed in the lower base and then rises above the others while forming a decreasing pentagon, all as with the model represented in Fig. 66: but the penetagonal film decreases much more rapidly, then vanishes and the system instantaneously undergoes a sharp change assuming a bizarre disposition which is difficult to represent in an exact manner by a drawing in perspective, but of which I shall nevertheless try to give an idea. On the two bases, two identical reentrant assemblages are supported, which are composed of five curved films, and of which one is represented in projection onto the plan of the base of Fig. 75: we see that in each of them, there is a pentagonal film two quadrangular films, and two triangular ones; these two assemblages are linked to each other by the films which emanate from five lateral edges of the prism, and by two other intermediate films also directed along the length of the wire model, and emanating from the liquid edges ab and bc of one of the same assemblages to abut on the homologous liquid edges of the other. I do not need to mention here that the same thing would occur also if, instead of surpassing the indicated limit, wed confined ourselves to only reaching it, that is, if we gave the prism the height which would precisely correspond to the simple annihilation of

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Figure 75 the pentagonal film in fact, there would be ten liquid edges meeting at the same point at the centre of the system and, consequently, equilibrium would be unstable. § 196. Though the oblique films must be considerably curved in the laminar systems of prisms with a greater number of sides, it seemed to me probable that, for each of these prisms, there must also be a limit of height, beyond which the system could not enclose a central polygonal film, and that this limit must be little different from that for the pentagonal prism. To verify it, I have first tried the wire hexagonal prism whose height was also almost 2! times the diameter of the circle inscribed in the base. To my great surprise, a central hexagonal film still formed, though much smaller than in the model considered in §186; but the system had undergone a modification which made the existence of this film possible: the points of the lateral solid edges from which the oblique liquid edges emanated (§191) were situated much farther from the vertices of the bases, so that things were arranged almost as if, in reality, the wire model had been shortened. In this disposition, the films emanating from the sides of the two bases remain adjacent to the solid lateral edges until they are at a sufficiently large distance from these, from which it follows that we have to consider the set as an imperfect laminar system; I say imperfect, because the films which emanate from all the sides of the same base are separated from each other and rendered independent by these edges in the parts which remain attached to the lateral solid edges. § 197. In prisms with the number oflateral faces exceeding six and in which, therefore, the angle between two adjacent faces is greater than 120 0 , the films remain attached to the lateral solid edges on a larger scale. For example, in a wire octagon in which the ratio of the height to the diameter of the circle inscribed in the base is almost the same as in the above models, the central octagonal film, instead of being very small is, on the contrary, very large and the two oblique liquid edges emanating from one of these vertices start being attached to the corresponding solid lateral edge at two points about one-sixth of the length of this edge distant from each other, and hence a little less than one-half of the diameter of the circle inscribed in the base; in this case, therefore, the films emanating from two homologous sides of the bases abandon the lateral solid edges to be directed towards the central octogonal film only by approaching the middle of the height of the wire model and, until then, they occupy the lateral faces of the prism and have a reasonably plane shape. In the wire hexagon of the previous section, the distance between the points where the two liquid edges are attached to the same lateral solid edge, emanating from one of the vertices of the central film, is about twice the diameter of the inscribed circle; in the wire

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octagon, it is, as we have seen, a little smaller than one-half of this diameter; in a wire heptagon, it is, which had to be expected, intermediate between these two values and equal to almost three-fourths of the same diameter. I have also tried a wire decagon, and in this, the distance in question is only one-sixth of the diameter. The thickness of metal wires has little influence on these facts; however, the distance between the points where oblique liquid edges are attached increases a little in skeleton figures in which the lateral edges are made of a very thin iron wire. But here some difficulties arise: the direction of the reentrant films is evidently regulated by the necessity for these films to have angles of 120 0 between them and, on the other hand, to be supported by the concave sides of the central polygonal film. In the models that we have considered these conditions seem to be equally fulfilled if the central polygonal film is smaller and if, at the same time, the films start to reenter at a lesser distance from the bases; we can therefore ask ourselves what determines the large stretch of the central film and the small distance between the points where oblique liquid edges are attached in the wire prisms with the number of sides greater than six. We say, from this point, that the reason lies in a minimum principle which we shall expound below (§209) and which all the laminar systems obey. § 198. I then constructed a wire octagon whose height was only one-third of the diameter of the inscribed circle. Then, since the given value was higher for the distance between the points of attachment in question, all these points would have to be at the vertices of the prism; but this was not so: the same points were still found to be at a certain distance from the vertices, and the distance between them was not greater than one-sixth of the diameter of the inscribed circle; also, the octagonal film increased still more. The same effect was produced in a heptagonal prism whose height was one-half of the diameter of the inscribed circle, that is, also less than the distance between the points of attachment evaluated previously with respect to prisms with this number of sides. In addition, this happens even in the hexagonal prism, since (§19l) the points where oblique liquid edges are attached are still situated at a small distance from the vertices in a wire model of this sort whose height is only 1~ times the diameter of the inscribed circle. § 199. The explanation of these latter facts results from what we said at the end of § 191: consider an octagonal or heptagonal wire prism, sufficiently high for the films emanating from the sides of the bases to occupy considerable portions oflateral faces and have a reasonably plane shape. At the places where these films leave the faces in question to make for the sides of the central polygonal film, they are, as we know, concave towards the exterior with respect to their width; hence, if we imagine the skeleton figure to be intersected by two planes perpendicular to its axis and passing through two series of points where the oblique liquid edges start on the lateral solid edges, these two planes will cut the films along arcs concave towards the exterior and, if we imagine these arcs to be solidified, the equilibrium of the system will not be disturbed. Hence, if we were to construct a skeleton figure having the height equal to the distance between the points where oblique liquid edges are attached on the same lateral solid edge and, if we were to give the curvature of the above-mentioned arcs to the iron wires which form the sides of the bases, then it is clear that the laminar system realized in this wire model would have

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its oblique liquid edges emanating exactly from the vertices; but with a wire model of this or lesser height, and in which the sides of the bases are straight, the condition relative to the transversal curvatures of the reentrant films and, consequently, to the sort of equilibrium of these films, can be evidently satisfied only if the points where oblique liquid edges are attached are placed at a certain distance from the vertices on the lateral solid edges. The facts that we have mentioned prove, as we see, what we have predicted at the end of § 197 concerning the possibility of equilibrium of a prism with some number of sides given with different dimensions of the central film and different distances of the points where oblique liquid edges are attached. §200. Ifwe compare the central polygonal film in the different systems that we have studied, we see that the curvature of their sides increases from the hexagonal film to the decagonal film, which is what should be added again to the facts established in § 186 in confirmation of the law relative to the angles at which the liquid edges meet at the same "liquid" point. The transversal curvature of the oblique films that make for the sides of the central polygonal film is related to the curvature of the same sides, and must be therefore less in the heptagonal prism than in the octagonal, and still less in the hexagonal; it is because of instability of the curvature in question in this latter prism, which gives, when its height is not too large, an almost perfect laminar system with its hexagonal film. §201. Reflecting upon the generation of laminar systems, I have asked myself if, at least in the hexagonal prism and with the model considered in § 196, we could not realize a system free of the central polygonal film when the wire model is removed from the glycerine solution, so that the axis of the prism might be horizontal instead of being vertical as in the previous experiments. I therefore gave the fork a disposition which permitted me to act thus and, in fact, succeeded completely. More than that, I obtained two different systems according as the wire model was removed so that two lateral edges emerged from the liquid simultaneously, or first one of them appeared and then simultaneously its two neighbours; these two systems are of the sort which is realized in a sufficiently high wire pentagon (§ 195), that is, those that are composed of two assemblages of oblique curved films linked to each other by other films directed along the length of the prism. The projection of one of these assemblages onto the plane of the base is represented in the first mode in Fig. 76, and in the second in Fig. 77. But this is not all. In order to produce the first of these two systems, it is required to complete the operation with a very slow tempo when we remove half of the wire model

Figure 76

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Figure 77 from the liquid; when we act without this precaution, a third system of yet another sort forms, an exact idea of which is hard to give by a description or drawing, despite its simplicity: it contains two curved hexagonal films supported respectively by one of their sides on one of those of the bases and directed obliquely towards the interior of the wire model; the other sides of these hexagons, as always, have concave curvature; the curved sides of each of the same hexagons are linked to the corresponding sides of the neighbouring base and to the homologous sides of the other hexagon by the curved films; finally, other films emanating from the lateral edges of the wire model abut on the liquid edges which unite these latter films pairwise. The sides of the bases by which the two hexagonal films are supported belong to the face of the prism which emerges from the liquid first. The heptagonal prism gives analogous results with a wire model having its dimensions in the same ratio. However, in the first place the three systems are imperfect in the sense that, in the first two of them, the films emanating from the sides of the bases remain adjacent to the lateral solid edges until a certain distance from the vertices is reached, while in the third, the films that start to go from the curved sides of one of the heptagonal films to the homologous sides of the other are attached to the lateral solid edges for the greatest part of their length, while assuming an appreciably plane shape on this scale; moreover, there is a new bizarreness, namely, that the system furnished in the second mode is unstable; hardly formed, it starts modifying spontaneously: the two assemblages situated near the bases elongate, first slowly, then ever more quickly, reach each other, and immediately the imperfect system with the heptagonal film in the middle appears. The projection of one of the assemblages of the system resulting from the first mode, onto the plane of the base is represented in Fig. 78; the projection relating to the second mode cannot be drawn, because the spontaneous modifications undergone by the system hamper satisfactory observation.

Figure 78

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The facts I have described show, in addition to what I reported at the end of § 193, that, with certain wire models, the results differ in accordance with the manner in which these models are removed from the glycerine solution. Presuming that the instability of the second system of the wire heptagon can be explained by this heptagon not having enough height, I have constructed another one of length three times the diameter of the inscribed circle; but I have gained nothing; moreover, the first two systems were more difficult to produce and almost always, the third one was obtained, that is, the one which contains two oblique heptagonal films emanating respectively from one of the sides of the bases. Finally, with a model still more elongated, we always have only this latter system. As to the octagonal prism, no matter what modificaticn the ratio between its length and the diameter of the inscribed circle underwent, it obstinately refused to give something other than the imperfect systems with the octagonal film in the middle, or with the two oblique octagonal films and this latter is also the unique one which is realized when the ratio is sufficiently large. §202. M. Van Rees, who has done me the honour of repeating my experiments in Holland, has found a very remarkable principle with respect to the laminar systems of wire prisms; he was kind enough to communicate it to me, and I am going to expound it here; however, this principle will be understood without difficulty only by those persons who have seen the systems to which it applies. The systems under consideration are of the kind whose projections are given in Figs. 75 and 78. In the systems of this sort, as we know, by each of the bases of the wire model a kind of a reentrant pyramid the vertex of which is formed by small liquid edges united to each other is supported, these edges being united either endwise while forming angles as in Figs. 75 and 77, or so that there are three of them meeting at the same point, as in Figs. 76 and 78; from each of these small edges there emanates, as we also know, a film parallel to the axis of the prism and due to be attached to the small homologous edge of the other pyramid with the other extremity. Here is the principle enunciated by M. Van Rees: If n denotes the number of lateral faces of the prism, then: 10. The number of small edges forming the vertex of each of two reentrant pyramids, and hence also the number of the longitudinal films going from small edges of one of the pyramids to those of the other is n - 3: 2 0 • The number of systems of this sort, realizable in the same wire model, is equal to that of the different open figures which can be formed with the n - 3 small edges, provided there are never more than three meeting at the same point. Thus, for the triangular prism, we have: n - 3 = 0 and we know, in fact, that in the laminar system of this wire model, the vertex of each of the reentrant pyramids is a simple point. For the quadrangular prism, n - 3 equals I, there is a unique small edge at the vertex of each pyramid, and hence a unique system possible of the sort we are considering. For the pentagonal prism, n - 3 is equal to 2, there are two small edges and, as it is possible to form only one open figure with these, namely, an angle (Fig. 75), there is also a unique system possible of the sort in question. For the hexagonal prism, the principle indicates three small edges and, therefore, the figures:

I\~A Hence, there are three systems; I have produced two of them (Figs. 76 and 77) and M.

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Van Rees has produced all three. For the heptagonal prism, the principle leads to the figures:

Hence there are four systems; my experiments have provided me with only two, and all four have been obtained by M. Van Rees. For the octagonal prism, there are at least thirteen figures and hence, thirteen systems; I have not succeeded in developing a single one; M. Van Rees has obtained many and believes that all can be realized, at the same time supposing that most of them are unstable. To promote the generation of the systems in question, M. Van Rees first obtains a system containing a polygonal film parallel to the bases in the middle of its height; then he dips a small part of the base of the wire model in the liquid and removes it; the plane film which will occupy this base and which thus encloses some air in the space bounded by it by the oblique films and by the polygonal film, rises on pushing this latter before it and determines, in the middle of the figure, a sort of closed film polyhedron whose lateral faces, have the same number that are convex towards the exterior, as those of the prism represented by the wire model; this done, he tears one of these lateral faces, and the resultant system is then that in which the small edges I have spoken of constitute a broken line all of whose angles are of the same sense. He therefore passes from this system to one of the others that the model admits, while blowing on one or other ofliquid edges parallel to the axis across the plane of one of the interior longitudinal films to which this edge belongs. He adds that sometimes the first system of this sort is obtained much more easily without realizing the interior film polyhedron first, by simply blowing on one of the edges of the polygonal film. The principle of M. Van Rees undoubtedly applies to all wire prisms whatever the number of their lateral faces; but I do not think that it is possible to advance the experimental verification beyond the octagonal prism, for which it is already difficult. §203. It is not only with the wire prisms that M. Van Rees has employed his partial dipping method; when, after having obtained the ordinary system of any wire model whatever, one of its faces is dipped to the extent of several mm, and then removed, the film that fills this face and which then rises among the others, always determines the formation of an interior film polyhedron. Generally, this polyhedron has the same number of faces as that represented by the wire model, and its faces are always more or less convex. We often obtain very beautiful results in this manner: for example, with the wire cube, the new system (Fig. 79) contains in the middle a film cube with slightly convex edges and faces which are attached by their edges to the films emanating from solid edges; also, with the wire tetrahedron, the new system presents a film tetrahedron with convex edges and faces in the middle, etc. These convexities are easily explained by the laws relating to the angles of the films and liquid edges. The films which constitute the interior polyhedra in question are not, evidently, of zero mean curvature; this is because one of their faces is in contact with a limited mass of air and not with free air. Below, I shall return to the same polyhedra, of which M. Lamarle has made a special study. M. Van Rees who, as we have seen, succeeds in modifying certain systems by blowing

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Figure 79 on them in a convenient manner, has found an interesting application of this procedure to the ordinary system of the wire cube (Fig. 59): in this system, the central quadrangular film is, as we know, parallel to two opposite faces of the cube; but, because of the symmetry of the wire cube it is evidently immaterial for equilibrium whether this parallelism occurs with respect to one pair of faces or another; the small film can, therefore, occupy three positions as well, and we understand that it is suffices to have a very trifling cause to be guided in its choice. Also, when we remove the model from the glycerine solution, we find the small film in question now parallel to the front and back faces, now parallel to the right and left, and sometimes even horizontally placed. We can make it pass at will, and many times in a row, from one of these three positions to ano.her; for that, it suffices to blow on one of these edges from the face of the skeleton figure at whose side this edge is placed: we then see the small film shrink in the direction of blowing, reduce to a simple line, and then reproduce itself in its new position. By a method that I shall indicate, a modification of another sort, and a fairly curious one at that, is determined in the systems of certain wire models: if, after having realized the ordinary system of the cube, the small central quadrangular film is torn, then the system immediately assumes quite a different, but also regular, disposition, though it has a void in its middle; we obtain a similar result, but with two voids, by tearing in the system of the octahedron (Fig. 63), first the upper quadrilateral and then the film that replaces the lower quadrilateral.

§4. Some Works of the Early Twentieth Century (Rad6, Douglas). In the present section, we give certain fragments from the famous papers of Rad6 [147] and Douglas [48]. The reader will have an opportunity to get

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acquainted with the solution of the two-dimensional Plateau problem in threedimensional Euclidean space. These results crown a whole series of investigations of many authors, and are a brilliant achievement on the long path of the development of the Plateau problem in its various statements.

T.Rado On the Plateau problem

§5 Minimal Surfaces in the Large IlL!. The PLATEAU problem, as considered for instance by H.A. SCHWARZ in his classical investigations*, calls for a minimal surface M which is bounded by a given Jordan curve r* and which is free of singularities in its interior. The question naturally arises whether this supplementary condition can be complied with ifr* is very complicated. The very particular cases which have been dealt with in older literature certainly cannot be considered as representative of what might be expected in the case of a general JORDAN curve. At any rate, that supplementary condition has been dropped, at least for the time being, in the modern general investigations, and the existence of a solution free of singularities has been established only for certain special classes of curves. 111.2. If the solution is permitted to have singularities, it is clearly necessary to specify the nature of these singularities. For instance, if the solution were permitted to have edges, then every polyhedron, bounded by a given polygon r*, would have to be considered as a solution of the problem of PLATEAU for the contour r*. Or consider a uniqueness theorem, for instance the theorem stated by H.A. SCHWARZ that the solution of the problem of PLATEAU is unique if the given contour is a skew quadrilateralt . H.A. SCHWARZ had only solutions in mind which are free of singularities. We shall see (111.12) that the theorem remains valid even if the solution is permitted to have certain singularities. On the other hand, the theorem breaks down if the specifications as to permissible singularities are too liberal, as follows from the remark made above. 111.3. The specifications as to the permissible singularities are by no means uniform in the recent literature. In other words, different authors do not always consider the same problem. We first consider the most liberal specifications actually used in the literature. Given a continuous surface S:r = r (u, v), (u, v) in R, where R denotes a JORDAN region bounded by a JORDAN curve r, and given also a JORDAN curve r* in the xyz-space, we shall say that S is a minimal surface (of the type of the circular disc) bounded by r* if the following conditions are satisfied* . • Gesammelte Mathematische Abhandlungen Vol. I. t Gesammelte Mathematische Abhandlungen Vol. I, p. 111. T. RAD6: Contributions to the theory of minimal surfaces. Acta Litt. Sci. Szeged Vol. 6 (1932) pp. 1-20.

*

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1. The equations of Scarry r in a topological way into r*. 2. For every point (uo' Yo) interior to r there exists a neighbourhood Vo and a one-to-~e continuous transformation a = ao(u, v), {3 = {30 (u, v) ofVo into some region Vo of the a{3-plane such that the components x, y, z of r as functions of a, {3 are harmonic in Vo and satisfy there the relations

E = G, F = 0, where E = ri, F = rx r~, G = r~. Parameters a, (3 with this property will then be called local typical parameters for the neighbourhood of(uo' Yo). 111.4. IfEG - p2 > 0 at the point (a o' (30) into which (uo' Yo) is carried, then the corresponding point of S will be called a regular point. The neighbourhood on S, of a regular point is a minimal surface in the sense of differential geometry. If EG - p2 = 0 at (ao' (30)' then we have, on account of E = G, p2 = 0, x", = y", = z", = x~ = y~ = z~ = 0 at (a o' (30). Suppose that all the partial derivatives of x, y, z with respect to a, {3 vanish at (ao' (30) up to and including a certain order n 2:: 1, while at least one of the derivatives of order n + 1 is different from zero. Then the point corresponding to (a o' (3o) on S will be called a branch-point of order n. The preceding notions are independent of the special choice of the local typical parameters a, {3. If S has only regular points, then S as a whole is a minimal surface in the sense of differential geometry, and will be called a regular minimal surface. 111.5. We shall now state the problems which will be discussed in this report. First we have the following statements of the problem of PLATEAU, which have been considered actually in the literature. PROBLEM Pl. Given, in the xyz-space, a JORDAN curve r*, to determine a minimal surface, of the type of the circular disc, bounded by r* (the term minimal surface being used in the sense of 111.3). PROBLEM Pz. Solve problem Plunder the supplementary condition that the solution admits of a representation S:r = r(u, v), u 2 + v2 ::5 1, where the components x(u, v), y(u, v), z(u, v) of r (u, v) are continuous for u 2 + v2 ::5 1, harmonic for u 2 + v2 < 1, and satisfy for u 2 + v2 < I the equations E = G, F = o. Furthermore, the equations x = x(u, v), y = y(u, v), z = z(u, v) are required to carry u 2 + v2 = 1 topologically into the given JORDAN curve r*. PROBLEM P J • Solve problem P 2 under the supplementary condition that the functions x(u, v), y(u, v), z(u, v) admit of a representation of the form x = RJ(cf>2 - 'lr2)dw, Y = Rji(cf>2 + 'lr2)dw, z = Rj2cf>'lrdw, where cf> and 'Ir denote single-valued analytical functions of w = u T iv in Iwl < 1. PROBLEM P 4• Solve problem P 3 under the supplementary condition that cf>, 'Ir do not have any common zero in Iwl < 1.

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PROBLEM P s. Suppose that the orthogonal projection of the given JORDAN curve r* upon the xy-plane is a simply covered JORDAN curve r. Denote by R the JORDAN region bounded by r. Solve problem Plunder the supplementary condition that the solution admits of a representation S: z = z(x, y), (x, y) in R, where z(x, y) is single-valued and continuous in R and analytic in the interior of

R. Besides the problem of PLATEAU, we shall have to consider the problem of the least area, which requires the determination of a continuous surface S bounded by a given JORDAN curve r*, such that the LEBESGUE area U (S) of S is a minimum when compared with the areas of all continuous surfaces bounded by r*, it being understood that only surfaces of the type of the circular disc are considered*• Then comes the simultaneous problem, which requires the determination of a common solution of the problem of PLATEAU and of the problem of the least areat . According to the different statements of the problem of PLATEAU, we have, strictly speaking, a number of simultaneous problems. As a matter of fact, only the statements P 2 and P 5 of the problem of PLATEAU have actually been used in this connection. Anyone of the preceding problems gives rise to the question as to whether or not the solution is unique. Furthermore, there arises the question as to whether or not some of these problems are equivalent. The existence theorems concerned with the problems listed above will be discussed in subsequent Chapters. The other questions raised in this section will be considered in the present chapter. Sections III.6 to III.16 contain a number of special facts which are then coordinated in the sections III.17 to II1.20. II1.6. The following definitions prove useful for the sequel. Let there be given, in a JORDAN region R of the uv-plane, a continuous function g(u, v). The g(u, v) will be called a generalized harmonic function in R if for every interior point (u o' vo) ofR the following condition is satisfied. There exists a neighbourhood Vo of (uo' vo) and a topological transformation u = U o (0:, (3), v = Vo (0:, (3) ofVo into some region '10 of an 0: (3-plane, such that the function g [uo(o:, (3), Vo (0:, (3)] of 0:, {3 is harmonic in '10. Variables 0:, {3 with this property will be called local typical variables for the point (u o' v0). Suppose that g(u, v) is a generalized harmonic function in R and that g(u, v) vanishes at an interior point (u o' vo) of R. Let (0: 0, (3o) be the image of (uo' vo) under the transformation u = U o (0: 0, (30)' v = vo(o:, (3), where 0:, {3 are local typical variables for (uo' vo). Suppose that all the partial derivatives of g with respect to 0:, {3 vanish at (0: 0, (30) up to and including a certain order (n - 1) ~ 0 while at least one of the partial derivatives of order n is different from zero. Then * Surfaces of different topological types will only be considered at the end of Chapter VI. t This problem has been called the problem of PLATEAU by LEBESGUE: Integrale, longeur, aire. Ann. Mat. Pura Appl. Vol. 7 (1902) pp. 231-359.

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(uo' vo) will be called a zero of order n of g(u, v). This notion is independent of the particular choice of the lcoal typical variables ex, {3. 111.7. If gl (u, v), g2(U, v) are generalized harmonic functions in R, then gl + g2 generally is not a generalized harmonic function. On the other hand, several important properties of harmonic functions remain valid for generalized harmonic functions. The property expressed by the principle of maximum and minimum clearly remains valid. The following lemma is an immediate consequence of well-known properties of harmonic functions which remain valid for generalized harmonic functions. LEMMA *. Let g(u, v) be a generalized harmonic function in a (simply connected) JORDAN region R. Suppose that g(u, v) has a zero (uo' vo) of order n ~ 1 in the interior ofR. Then g(u, v) vanishes in at least 2n distinct points on the boundary ofR. 111.8 Consider now a surface S:r = r(u, v), (u, v) in R that is a minimal surface (in the sense defined in III.3) bounded by a JORDAN curve r*. If ex, {3 are local typical parameters (see 111.3) for an interior point (uo' vo) of R, then the components x, y, z of r as functions of ex, {3 are harmonic functions. Hence, if a, b, c, d are any four constants, then ax + by + cz + d is also a harmonic function of ex, (3. Thus if a, b, c, d are any four constants, then the function ax(u, v) + by(u, v) + cz(u, v) + d is a generalized harmonic function in R. The following theorems are immediate consequences of this remark, on account of the facts referred to in 111.7. 1. If a convex region K in the xyz-space contains the boundary r* of a minimal surface (see 111.3) then the whole surface is contained in K**. 2. The tangent plane, at a regular point (see 111.4) of the minimal surface, interacts the boundary curve in at least four distinct pointst. 3. Every plane passing through a branch-point of order n (see 111.4) of the minimal surface intersects the boundary curve r* in at least 2(n + 1) distinct pointstt. III. 9. The last theorem permits us to exclude the possibility of branch-points in certain cases. Suppose there exists, in the xyz-space, a straight line 1 such that no plane through 1 intersects the boundary curve r* in more than two distinct points. Then the minimal surface cannot have branch-points, as follows immediately from theorem 3 in 111.8. The assumption is satisfied, for instance, if * T. RAD6: Contributions to the theory of minimal surfaces. Acta Litt. Sci. Szeged Vol. 6 1932) p. 10, where the lemma is stated for n = 2. • ** The reviewer learned about this theorem from L. FEJER.

The theorem is also true for surfaces with negative curvature. This fact played an important role in the work of S. BERNSTEIN on partial differential equations of the elliptic type. See for references L. LICHTENSTEIN: Neuere Entwicklung usw. Enzyklopadie der math. Vol. 2(3) pp. 1277-1334. tt See T. RAD6: The problem of the least area and the problem of PLATEAU. Math. Z. Vol. 32 (1930) p. 794, where the theorem is stated for n = 1.

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r* has a simply covered star-shaped JORDAN curve as its parallel or central projection upon some plane. If the projection has the stronger property of being convex, then a much stronger conclusion can be drawn. Suppose that the parallel projection of r* upon some plane is a simply covered convex curve. Choose a plane perpendicular to the direction of projection for xy-plane. Then the orthogonal projection of r* upon the xy-plane is again a simply covered convex curve which we shall call r. Let S: x = x(u, v) y = y(u, v), z = z(u, v), (u, v) in R be the minimal surface under consideration. From theorem 3 in 111.8 it follows that S has no branch points. From theorem 2 in 111.8 it follows that S has no tangent plane perpendicular to the xy-plane. From this it follows that S has a simply covered xy-projection in the small. Hence the equations x = x(u, v), y = y(u, v) define a transformation with the following properties. 1. The transformation is one-to-one and continuous in the vicinity of every interior point (u o' vo) of R. 2. The boundary of R is carried in a one-to-one and continuous way into the JORDAN curve r. On account of the so-called monodromy theorem in topology *, it follows then that the transformation x = x(u, v), y = y(u, v), (u, v) in R carries R topologically into the JORDAN region bounded by r. Hence u, v can be expressed as single-valued continuous functions of x, y in R, and there follows then for the minimal surface S a representation S : z = z(x, y), (x, y) in or on r, where z(x, y) is single-valued and continuous in and on r. Since S has no branchpoints, S is a minimal surface in the sense of differential geometry. Hence (see 11.16) z(x, y) is analytic in the interior of r and satisfies there the partial differential equation (1 + q2)r - 2pqs + (1 + p2)t =

o.

(3.1)

Similar conclusions may be obtained if the boundary curve r* is supposed to have a simply covered convex curve as its central projection upon some plane. 111.10. Summing up, we have the following resultst . Let S be a minimal surface (in the sence of 111.3) bounded by a JORDAN curve r*. If r* has a simply covered star-shaped JORDAN curve r as its parallel or central projection upon some plane, then S has no branch-points, that is to say S is a minimal surface in the sense of differential geometry. If the projection r is convex, then S does not intersect itself even in the large. If the orthogonal projection ofr* upon the xy-plane is a simply covered convex curve r, then S can be represented in the form S : z = z(x, y), (x, y) in or on r, where z(x, y) is single-valued and continuous in and on r, and satisfies in the interior of r the partial differential equation (3.1) . • See, for instance, KEREKJART6: Vorlesungen iiber Topologie I, p. 175. t See T. RAD6: The problem of the least area and the problem of PLATEAU. Math. Z. Vol. 32 (1930) pp. 763-796. - T. RAD6: Contributions to the theory of minimal surfaces. Acta Litt. Sci. Szeged Vol. 6 (1932) pp. 1-20.

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111.11. We are going to consider now certain uniqueness theorems. A first important fact in this connection is the uniqueness theorem for the partial differntial equation (3.1). Let there be given, on a JORDAN curve r in the xyplane, a continuous boundary function 4> (P) of the point P varying on r. If then Zl(X, y), Z2(X, y) are solutions of (3.1) which both reduce to Ql (P) on r, then Zl(X, y) z2(x, y) in the whole interior of r*. Denote then by r* the JORDAN curve, in xyz-space determined by the equation Z = 4>(P). The above uniqueness theorem asserts that r* cannot bound more than one minimal surface that has a simply covered xy-projection. This statement is rather unsatisfactory; indeed, we shall see (111.17) that the boundary curve of a minimal surface may very well have a simply covered xy-projection, while the minimal surface itself does not have this property. On the other hand, if the xy-projection of the boundary curve r* is convex, then every minimal surface bounded by r* also has a simply-covered xy-projection, on account of 111.10. A similar argument holds in case r* is known to have a simply covered convex curve as its central projection. This results in the following uniqueness theoremt .

=

If a JORDAN curve r * has a simply covered convex curve as its parallel or central projection upon some plane, then r * cannot bound more than one minimal surface (this term being used in the general sense defined in IIl.3). 111.12. The assumptions of the preceding uniqueness theorem are obviously satisfied if r* is a skew quadrilateral. For this case, the uniqueness theorem has been stated without proof by H.A. SCHWARZ. For the purpose of an application to be made later on we mention the following consequence of the uniqueness theorem. Suppose a JORDAN curve r* is invariant under reflection in a certain plane p. Every minimal surface, bounded by r*, is carried by the reflection into a minimal surface bounded by r*. Hence, if it is known that r* bounds just one minimal surface S, then it follows that S is also invariant under the reflection. Repeated application of this remark to the case when r* consists of the edges AB, BC, CD, DA ofa regular tetrahedron with vertices A, B, C, D, leads to the result that the minimal surface bounded by r* passes through the centre of the tetrahedron (this fact has been verified by SCHWARZ by using the explicit formulae for the surface). 111.13. Let us consider now the relation between the problem of the least area and the simultaneous problem (see III. 5). If a solution S of the problem of the least area satisfies the assumptions made in the classical Calculus of Variations, then S is a minimal surface (see 11.5). Without those assumptions this conclusion * See, also for references, the beautiful treatment of this theorem and of related subjects by E. HOPF: Elementare Bemerkungen tiber die Liisungen partieller Differentialgleichungen zweiter Ordung )!om elliptischen Typhus. S.-B. preu {3. Akad. Wiss. 1927 pp. 147-152. See also A. HAAR: Uber reguliire Variations-probleme. Acta Litt. Sci. Szeged Vol. 3 (1927) pp. 224-234. . t T. RAD6: Contributions to the theory of minimal surfaces. Acta Litt. Sci. Szeged Vol. 6 (1932) pp. 1-20.

129

SURVEY OF MINIMAL SURFACE THEORY

does not in general hold. The following example is a slight modification of one given by LEBESGUE in his Thesis*. Let the given JORDAN curve r* coincide with the circle r* : x2 + y2

= 1, z = o.

Then the area U (S) of any continuous surface S (of the type of the circular disc) bounded by r* is ~ 1r. Consider then the surface y = 2n(r - tfsin cf>, y= 0

z = 0 for t ~ r ~ 1, z = (1 - 4r2)n for O~r~

t,

where r, cf> are polar coordinates in the uv-p1ane, and n is a given positive integer. Using the relations u = r cos cf>, v = r sin cf>, we have the equations of S appearing in the form S: x = x(u, v), y = y(u, v), z = z(u, v), u 2 + v2 ::5 1, where x(u, v), y(u, v), z(u, v) are easily seen to have continuous partial derivatives up to and including the order n - 1. S consists of the simply covered disc x2 + y2 ::5 1, z = 0 and of the spine x = 0, y = 0,0 ::5 z ::5 1. The area U (S) is found to be equal to 7r by computing the integral H(EG - F2)t. Hence the area ofS is a minimum. Still, S is not a minimal surface, not even in the general sense defined in III.3. Indeed, a minimal surface is contained in every convex region that contains its boundary curve (see III.8), and S obviously does not satisfy this condition. Instead of using one spine as above, we can disfigure any given surface S by putting on it any finite number of spines, without changing its area. We shall see (Chapter VI) that the problem of the least area is solvable for every JORDAN curve. From these two facts it follows that the problem of the least area has infinitely many solutions for every JORDAN curve r *, and that a surface S which solves the problem is not in general a minimal surface. 111.14. Thus the solution of the problem of the least area does not imply the solution of the problem of PLATEAU. Neither does the solution of the problem of PLATEA U imply the solution of the problem of the least area; in other words the area of the minimal surface, bounded by a given JORDAN curve r*, is not necessarily a minimum. This fact had been recognized at the earliest stage of the theory. For the case of doubly connected minimal surfaces bounded by two given curves, the catenoids offer simple examples of the lack of the minimizing propertyt. For the cast of minimal surfaces of the type of the circular disc, bounded by a given curve, H.A. SCHWARZ obtained very general examples in the following way*. * Integrale, longuere, aire. Ann. Mat. pura appl. Vol. 7 (1902) pp 231-359. t See, also for references, the beautiful Chapter IV in the little book ofG.A. BLISS: Calculus of Variations (No.1 of the Cams Mathematical Monographs). Gesammelte Mathematische Abhandlungen Vol. 1 pp. 151-167 and 223-269.

*

130

THE PLATEAU PROBLEM: PART ONE

Consider, in the u + iv = w plane, a JORDAN region R bounded by an analytic JORDAN curve. Denote by u(w) a function which is analytic and different from zero in R. Then the equations w

X

=

RJ (I

- w2) Jt(w) dw,

w

y

= RJi (I

+ wZ) Jt(w) dw,

(3.2)

w Z

= RJ2wJt(w) dw,

where w varies in R, define a regular minimal surface'. The area of this surface certainly is not a minimum if the inequality (2.4) in 11.7 is not satisfied. The lefthand side of that inequality reduces in the present case to

8A J J[Au + Av. I- +( u + v) 2

R

2

2

2

2 2

]

du dv

(3.3)

Hence the area of the minimal surface certainly is not a minimum if this integral can be made negative by substituting a function A (u, v) that has continuous partial derivatives of the first order in R and that vanishes on the boundary of R. This criterion, curiously enough, does not depend upon the function Jt(w) which determines the minimal surface; the criterion is concerned solely with the region that R. SCHWARZ based the discussion of this situation in the study of the characteristic values of a certain partial differential equation". One of the results he obtained states that if the region R contains the unit circle u 2 + v2 :5 1 in its interior, then the integral (3.3) can be made negative. On account of the geometrical meaningt of the variable w in the formulas (3.2), this assumption concerning R means that the spherical image of the minimal surface defined by (3.2) completely covers half the unit sphere. The result of SCHWARZ can also be obtained in the following elementary way*. Let r be a positive parameter and define a function A (u, v; r) by

A(U,

V;

u2 +v-r r) = u2 + V + r for 0 ~ u2 + v~r.

Put J(r) =

u2

H[A~ + A~ - (1 + :;2+ vI ] du dv

+ v
* The formulas (3.2) are obtained by choosing>. (w) = w in (2.28). **Gesammelte Mathematische Abhandlungen Vol. 1 pp. 241-269. t w is the stereographic projection of the spherical image of the surface. See DARBOUX: Theorie generale des surfaces Vol. I pp. 347-348. T. RADO: Contributions to the theory of minimal surfaces. Acta Litt. Sci. Szeged Vol. 6 (1932) pp. 1-20.

*

SURVEY OF MINIMAL SURFACE THEORY

131

Partial integration gives J(r) =

-JJA [~A + (J + :; +1If ] du dv u2 + 11
Actual computation shows that the function A (u, v; 1) satisfies the equation

~A+ Hence J(l) =

8A =0 (J + u2 + 1If

o. We want now to determine ]'(1). Changing to new variables u

v

T

T

0=-,{3=-

we transform J(r) into a new integral taken over the fixed region 0 2 + (32 < 1. ]'(1) can then be computed by differentiating under the integral sign. The result

is

Since the integrand obviously is negative, it follows that J'(l) < O. From J( 1) = 0,

]'(1)< 0 it follows that we have a constant a> 1, such that J(r) < 0 for 1 < r < o. Suppose then that a JORDAN region R contains the disc u 2 + v2 ::51 in its interior. Then r can be determined such a way that 1 < r < 0 and that the disc

u 2 + v2 ~ r2 is interior to R. Define a function A (u, v) by the formulae A(U v) =

,

I

A(U, v; r)

0

in 0 fOT

~

u2 + 11·~ r2, u2 + 11 > r2.

Then A (u, v) vanishes on the boundary of R and makes the integral (3.3) negative. The first partial derivatives of A (u, v) are discontinuous on u 2 + v2 = r2; this edge however can be rounded off by a familiar process. It is thus proved that ifR contains the disc u 2 + v2 ::5 1 in its interior, then the formulae (3.2) define, for every choice of the analytic function Il(w) a minimal surface the area of which is not a minimum. 111.15. In order to obtain a clear-cut example, the function Il(w) in (3.2) should be chosen in such a way that the resulting minimal surface is bounded by a JORDAN curve. It can easily be shown that for Il(w) == i this condition is satisfied, provided R is a disc u 2 + v2 ::5 r2 with r < ..[3. Then the following explicit example is obtained*. The equations * T. RAD6: Contributions to the theory of minimal surfaces. Acta

(1932) pp. 1-20.

Litt. Sci. Szeged Vol. 6

132

THE PLATEAU PROBLEM: PART ONE

x = u +

UV2 -

y = -

V -

z = u2

-

U 2V

t

U\

+ t

V\

}

U2

+

V2 ~

r2, 1
.J3,

(3.4)

v 2,

define a minimal surface S bounded by ajORDAN curve, such that the area of Sis .. * not a mtmmum . 111.16. The area of the minimal surface S given by (3.4) is clearly finite. On account of a general existence theorem to be considered in Chapter VI, it follows that the boundary curve of S bounds a minimal surface S* the area of which is a minimum. Then S* is not identical to S, because the area ofS is not a minimum. Hence the boundary curve of S bounds at least two distinct minimal surfaces. The equations of the boundary curve in terms of polar coordinates r, 8, where u = r cos 8, v = r sin 8, are obtained in the form

x = r cos 8 - t r3 cos 3 8, } y=rsin8-tr 3 sin38, O~8<21r.

(3.5)

z = r2 cos 2 8. Thus we have the following example l : if 1 < r < .J3 , then the equations (3.5) define a JORDAN curve which bounds at least two distinct minimal surfaces of the type of the circular disc. The catenoids give explicit examples of distinct minimal surfaces with identical boundariest . It seems that no explicit example has yet been given for two minimal surfaces of the type of the circular disc and bounded by the same JORDAN curve. In the above example, one of the two minimal surfaces is given explicitly by the formulae (3.4), while the other is only known to exist*. At any rate, the solution of the problem of PLATEA U in general is not unique; on the other hand, the solution is unique for certain special classes of curves (see 111.11). It would be desirable to obtain more comprehensive information converning those properties of the given boundary curve which determine the number of solutions. 111.17. We shall now discuss briefly the statements of the problems PI to P 5' listed in 111.5. While problem PI simply calls for a minimal surface S (of the type of the circle) bounded by a given JORDAN cirve r*, problems P 2 to P 5 require that S admits of a representation of a prescribed type. In the case of problem P 5'

• The formulas (3.4) define the so-called minimal surface of ENNEPER. See DARBOUX: Theorie generale des surfaces Vol. I pp. 372-376. Examples of a less elementary character have been given by H.A. SCHWARZ: Gesammelte Mathematische Abhandlungen Vol. I pp. 151-167 and 223-269. t See G.A. BLISS: Calculus of Variations (No. I of the Carns Mathematical Monographs). The same remark applies to an example due to N. WIENER. See J. DOUGLAS: Solution of the problem of PLATEAU. Trans. Amer. Math. Soc. Vol. 33 (1931) p. 269.

*

133

SURVEY OF MINIMAL SURFACE THEORY

this implies a restriction on r* also, namely the restriction r* admits of a simply covered xy-projection. In the course of general investigations on partial differential equations of elliptic type, S. BERNSTEIN* observed that a large class of problems, including as a special case our problem P 5' in general is not solvable. The following examplet shows the geometrical reasons for this fact for problem P 5' Take a regular tetrahedron, with verticles A, B, C, D, and let the xy-plane pass through A. B. C. Denote by r* the quadrilateral AB, BC, CD, DA. The r* has a simply covered xy-projection. Yet, problem P 5 does not have a solution for r*. There certainly exists a minimal surface S bounded by r*, as has already been proved by SCHWARZ. We also know (IIUI) that S is the only minimal surface (of the type of the disc) bounded by r*. S passes through the center 0 of the tetrahedoron (III. 12), and thus S contains two points, namely 0 and D, with the same xy-projection. Hence there exists a unique minimal surface bounded by r *, and this minimal surface does not satisfy the additional condition of having a simply covered xy-projection*. This example shows very clearly that the trouble comes from the fact that problem P 5 is a non-parametric problem , inasmuch as it calls for a surface represented by an equation of the form z = z(x, y), instead of asking for a surface in the general parametric form x

= x(u,

v), y

= y(u, v),

z

= z(u,

v).

IIUB. The problems PI to P 4 (see III.5) are statements of the problem of PLATEAU in the parametric form. Problems P 2' P 3' P 4 call, roughly speaking, for a minimal surface in terms of isothermic parameters in the large. The character of this condition can be well illustrated in the special case when r* is a plane curve. Suppose r* is in the xy-plane, and consider, for instance, a solution

s: x = x(u, v), y = y(u, v), z = z(u, v), u 2 + v2

:5

I

of problem P 2' Then z(u, v) = 0 on u 2 + v2 = I. As z(u, v) is harmonic, it follows that z(u, v) ;: O. The conditions E = G, F = 0 reduces therefore to x~ + y~

= x; + y;,

XUXy

+

YuYy

= O.

This implies that x(u, v) and y(u, v) are conjugate harmonic functions, that is * S. BERNSTEIN: Sur les equations du Calcul des Variations. Ann. Ecole norm. Vol. 29 (1912) pg. 431-485. See in particular pp. 484-485. t T. RADO: Contributions to the theory of minimal surfaces. Acta. Litt. Sci. Szeged Vol. 6 (1932) pp. 1-20. The uniqueness is absolutely essential for the conclusion that problem P 5 is not solvable. For this reason, several examples presented in the literature are incomplete.

*

l34

THE PLATEAU PROBLEM: PART ONE

to say the real and imaginary parts of an analytic function f (w) of w = u + iv. The statement of problem P 2 requires that 1. f(w) is analytic for Iwl < 1, and 2. f(w) is continuous for Iwl ~ 1, and 3. the equation r = f{w) carries Iwl = 1 topologically into the JORDAN curve r* of the r = x + iy plane. On account of a classical theorem of DARBOUX (see V. 19), this situation implies that the equation r = f{w) carries Iwl < 1 in a one-to-one and conformal fashion into the interior of r*. Hence problem P 2 requires the mapping of the JORDAN region, bounded by r*, to be one-to-one and continuous onto the unit circle Iwl ~ 1 and conformal on its interior. The same holds obviously for problems P 3 and P 4 • While problems P 2' P 3' P 4 reduce, in the case of a curve situated in the xy-place, to one of the fundamental problems in conformal mapping; problem P, simply is trivial in this case. Indeed, ifR denotes the JORDAN region bounded by r*, then the surface S:x

= x,

y

= y, z = 0,

(x, y) in R

clearly solves problem P ,. It might be inferred from these remarks that problem P, is easier to solve than problem P 2. Curiously enough, nobody has ever tried to take advantage of this possibility. The problem of PLATEAU, in the parametric form, has always been asked in one of the statements P 2' P 3' P 4· which also require the mapping of the minimal surface to be conformal onto the unit circle in the large. The question naturally arises whether or not every solution of problem P, is also a solution of problem P 2. This amounts to the question whether or not a minimal surface (in the general sense of III. 3) bounded by a JORDAN curve r* always admits of a representation as specified by the statement of problem P 2. It seems that the . methods developed for dealing with the OSGOODCARATHEODORY theoremt will permit, after proper adjustments, the answering of this question in the affirmative*. III. 19 . Consider now a solution S:x

-----

= x(u,

v), y

= y(u, v), z = z(u, v), u 2

+ v2 ~ 1

* Or in the even stricter form based on the formulas (3.2). t We mean the following theorem: if the interior ofa JORDAN curve is mapped in a one-toone and conformal fashion onto the interior of the unit circle, then the map remains continuous and one-to-one on the boundaries. See for instance CARATHEODORY: Conformal representation (Cambridge Tracts 28). This program has been carried out in a joint paper by E.F. BECKENBACK and T. RADO: Subharmonic functions and minimal surfaces. To appear in Trans. Amer. Math. Soc.

*

SURVEY OF MINIMAL SURFACE THEORY

135

of problem P3 or P 4 (see 111.5). We then have the equations Xu -

ix" = 4>2 - '1'2,

)

Y u ~ .iy" _= i(4)2 + '1'2), Zu

IZ" -

(3.6)

24>'1'.

The condition that EG - F2 = 0 for an interior point (uo' v0) is then readily found to be equivalent to 4> = 0, 'I' = 0 for w = w0 = U o + ivo. Since problem P 4 requires that 4> and 'I' have no common zeros, we see that in problem P4 the solution is not permitted to have branch-points. It is easily seen that if a solution of problem P 2 does not have branch-points, then it admits of a representation as required in problem P 4 • In other words, if we add the condition EG - F2 > 0 for u 2 + v2 < 1 in the statement of problem P 2, then we obtain problem P 4, that is to say the classical statement of the problem of PLATEAU in parametric form *. The question arises whether or not problem P 4 is always possible. The impossibility, in general, of the problem would be demonstrated by exhibiting a single JORDAN curve r* for which it could be proved that every minimal surface (of the type of the circular disc) bounded by r* necessarily has branchpoints. Such a curve has not yet been exhibited; the conjecture that any knotted JORDAN curve would serve the purpose can readily be refuted by examples of knotted JORDAN curves which do bound minimal surfaces (of the type of the circular disc) free of branch-points. More explicitly: there exist knotted JORDAN curves for which the classical problem P 4 is possible, and no JORDAN curve is known at present for which problem P 4 is impossible t . Combining the existence theorem in Chapter V with the theorems in 111.10, we obtain existence theorems for the classical problem P 4 which seem to be the most general known at present. For instance: if a JORDAN curve r* has a simply covered star-shaped curve as its parallel or central projection upon some plane, then problem P 4 is solvable for r*. 111.20. While problem P 4 excludes branch-points altogether, and while problem P 2 does not imply any restriction as to branch-points, the geometrical interpretation of problem P 3 is less clear-cut. Differentiating again the equations (3.6), we obtain

~u - ~v

= 2(4)4>' - '1''1''),

Yuu - iyuv

= 2i(4)4>' + '1''1''),

} (3.7)

zuu - izuv = 2(4)'1'' + 4>''1'). From (3.6) and (3.7) it follows readily: if, for a solution of problem P 3' we have , * See H.A. SCHWARZ: Gesammelte Mathematische Abhandlungen Vol. 1. t Cf. VI.35.

136

THE PLATEAU PROBLEM: PART ONE

EG - F2 = 0 at an interior point U o + ivo = w o' then the partial derivatives of the first and second order of x(u, v), y(u, v), z(u, v) all vanish at that point. In other words: the branch-points of a solution of problem P J are at least of order 2 (see 111.4 for the definition). On the other hand, examples show that problem P 2 may have solutions with branch-points of order one. Thus it follows that the problems P 2 and P J are not equivalent. Problem P 3 will be seen to be solvable for every JORDAN curve that is not knotted (Chapter V). Again, it is not known at the present time whether there do or do not exist curves for which problem P 3 is impossible. III.21. We have seen (1I1.l8) that if the given JORDAN curve r* is situated in the xy-plane, then the problem of PLATEA U in parametric form (in anyone of the statements P 2, P 3, P 4 of 111.5) reduces to the problem of mapping the JORDAN region bounded by r* in a one-to-one and continuous fashion, and in the interior, in conformal fashion onto the unit circle u 2 + v2 ::5 1. This situation played an important role in the theory. The most direct illustration is given by the case when r* is a polygon. One of the earliest ideas for dealing with the problem of the conformal mapping of the unit circle onto the region bounded by a plane polygon was based on the principle of symmetry and led to the socalled formulas of SCHWARZ and CHRISTOFFEL. The method used by SCHWARZ in his classical investigations on the problem of PLATEAU is clearly a generalization to 3-space of the plane method'. In a general way, the reader will find, in the Chapters IV, V, VI dealing with the existence theorems, many instances where the theory of the conformal mapping of plane regions clearly served as a model for the development of the theory of the problem of PLATEAU. An important remark, due to J. DOUGLAS, should be mentioned here. Since the problem ofOSGOOD-CARATHEODORY (conformal mapping ofa plane JORDAN region upon the circle) is included in problem P 2, it follows that if we have a method of solution of problem P 2 which does not make use of the solution of that problem, then we have a simultaneous solution of the problem of OSGOOD-CARATHEODORY and of the problem of PLATEAU. J. DOUGLAS ~mphasizes the fact that this is the case with his own method( III.22. Instead of considering only existence theorems, the relation between conformal mapping of plane regions, that is to say, analytic functions of a complex variable, on the one hand and minimal surfaces on the other, might be discussed on account of its own intrinsic interest. The theory of minimal surfaces appears then as a generalization of the theory of analytic functions of a complex variable. While Chapters IV, V, VI will review numerous facts and methods • Compare, for instance, the following two papers of H.A. SCHWARZ: Uber einige Abbildungsaufgaben. Gesammelte Mathematische Abhandlungen Vol. 2 pp. 65-83 and Bestimmung einer speziellen Minimalflache. Gesammelte Mathematische Abhandlungen Vol. I pp. 6-125. See also DARBOUX: Theorie generale des surfaces Vol. I pp. 490-601. t See J. DOUGLAS: Solution of the problem of PLATEAU. Trans. Amer. Math. Soc. Vol. 33 (1931) pp. 263-321.

SURVEY OF MINIMAL SURFACE THEORY

137

which might be interpreted from this point of view, it might be useful to present to the reader some specific illustrations. III.23. If f (w) = x(u, v) + iy(u, v) is an analytic function of w for Iwl < 1, then x" = yv' x., = - Yu (CAUCHY-RIEMANN equations). from these equations it follows that (3.8) Conversely, it follows from (3.8) that either y is conjugate harmonic to x, or x is conjugate harmonic to y. Let us call two harmonic functions related by (3.8) a couple oj conjugate harmonic Junctions.

On the other hand, the theorem of WEIERSTRASS (see 11.17) leads one to consider triples of harmonic functions related by the equations

We shall say that x, y, z form a triple oj conjugate harmonic Junctions. We shall review now a few facts which develop further this analogy. 111.24. Suppose f (w) is analytic in Iwl < 1 and even on Iwl = 1, for the sake of simplicity. Put f'(w) = g(w). Then the area of the image of Iwl ::;; 1 by f(w) is given by 127r

U = f f Ig(re iB )I2r dr de, o 0

(3.10)

while the length L of the image of Iwl = 1 is 2".

L= flg(eiB)lde. o

(3.11)

If f (w) gave a simply covered image of Iwl ::;; I, then we could assert, on account of the isoperimetric inequality, that U ~t1TU,

or, on account of (3.10) and (3.11), that (3.12) CARLEMAN* proved that (3.12) holds regardless of whether f(w) gives a * Zur Theorie der Minimalfliichen. Math. Z. Vol. 9 (1921) pp. 154-160.

138

THE PLATEAU PROBLEM: PART ONE

simply covered image. From this he inferred that between the area A of a minimal surface and the length L of its boundary curve the isoperimetric inequality also holds. To prove this*, suppose, to simplify the discussion, that the minimal surface is given by

S : x = x(u, v), y = y(u, v), z = z(u, v), u2 + v2

~

1,

(3.13)

where x, y, z form a triple of conjugate harmonic functions (see II1.23) which remain analytic even on u 2 + v2 = 1. Then x, y, z are the real parts of analytic functions:

and if we put f; = gl' f2 = g2' f; = g3 and use the equation gy + g~ + g~ = 0 (see II.18), then the isoperimetric inequality is expressed by

t~

j 2rlgk(rei8Wrdrde~t.,.. [2{0

k-100

(t ~

k-l

Igk(e i8

)F) !de] 2

(3.14)

To prove (3.14), we observe that on account of the inequality of MINKOWSKIt we have

t E (Ylgk(ei8)lde)2~ [Y·(t k=l 0

0

E Igk(ei9W)tde]2.

k=l

(3.15)

Hence (3.14) follows immediately from (3.15) and (3.12). Some further discussion shows that the sign of equality in (3.14) holds if and only if the minimal surface reduces to a simply covered circular disc. If we suppose that the minimal surface has a minimum area, then the theorem of CARLEMAN is almost trivial, as has been observed by BLASCHKE:\:. Indeed, consider a cone consisting of the straight segments which connect a flxed point of the boundary curve with a variable point on that curve. Since cones are developable, the isoperimetric inequality holds for this cone, and hence a fortiori for the minimal surface, since the area of this latter is by assumption not greater than the area of the cone, while the boundary curve is the same for either surface. This remark of BLASCHKE shows that the point of the theorem of CARLEMAN is that the theorem is true even if the area of the minimal surface is not a minimum (cf. III.14). II1.25. Further inequalities between the area A of a minimal surface and the • We follow the simple proof given by E.F. BECHENBACH: The area and boundary of minimal surfaces. Ann. of Math: Vol. 33 (1932) pp. 658-664.

t See for instance POLYA-SZEGO: Aufgaben und Lehrsiltze Vol. 2 p. 14.

* T. CARLEMAN: Zur Theorie der Minimalflilchen. Math. Z. Vol. 9 (1921) p. 160.

SURVEY OF MINIMAL SURFACE THEORY

139

length L of its boundary curve have been obtained by BECKENBACH*. Suppose that the minimal surface is again given by the equations (3.13). Suppose also that at the origin we have E = 1 (that is to say, the linear ratio of magnification at the origin is unity). Then U

~ 11",

That is to say, A is at least equal to the area and L is at least equal to the perimeter of the unit circle. The sign of equality holds if and only if the minimal surface is a simply covered circular disc. These and similar theorems are proved by BECKENBACH by using FOURIER expansions. 111.26. A minimal surface S being again given by (3.13) where x, y, z are supposed to form a triple of conjugate harmonic functions, we shall call (x2 + y2 + Z2)! the norm of S and we shall write (x2 + y2 + Z2)! = lSI. Then lSI is the generalization of the absolute value of an analytic function f(w) of w. If f(w) is analytic, then log If(w)1 is a harmonic function ofu, v, and this accounts for many important facts in the theory of functions of a complex variable. While log lSI is not harmonic, it can be shown to be subharmonict . A function g (u, v) is subharmonic in a domain D iffor every point (uo' vo) ofD the inequality g(uo' vo) ~

21r

!". I g(uo + o

ecosS , Vo +

e sinS) dS

is satisfied for sufficiently small values of e*. If g has continuous' partial derivatives of the second order, then this condition is equivalent to ~g = g"u + ~ ~ 0 and log lSI is easily shown to satisfy this latter condition. It is sufficient to consider the situation in domains D where lSI > o. Put log lSI = g. Direct computation gives ~g = (r~ + rt,)r 2 - 2«rrj + (rry)

ISI 4

'

(3.16)

where r = r(u, v) is the vector equation of S. Since the components of r form a triple of conjugate harmonic functions, we have

* See E.F. BECKENBACH: The area and boundary of minimal surfaces. Ann. of Math.

Vol. 33 (1932) pp. 658-664.-The theorems in III.24 and III.25 also hold for surfaces of negative curvature, given in isothermal representation. See E.F. BECKENBACH and T. RAD6: Subhannonic functions and surfaces of negative curvature. To appear in Trans. Amer. Math. Soc. t III.26 to III.29 are taken from E.F. BECKENBACH and T. RAD6: Subharmonic functions and minimal surfaces. To appear in Trans. Amer. Math. Soc. See F. RIESZ: Sur les fonctions subhannoniques etc. Acta Math. Vol. 48 (1926) pp. 329-343.

*

140

THE PLATEAU PROBLEM: PART ONE

At those points where r~ = r; = 0 we have ~g = o. At those points where = r; > 0, ru and rv are different from zero and are perpendicular to each other. Using a unit vector ~ perpendicular to ru and rv we can write r~

where a, b, c are scalars. It follows that

Substituting in (3.16) we get

Thus ~g ~ 0 always. 111.27. The fact that log lSI is subharmonic makes it possible to extend a great number of theorems on analytic functions to minimal surfaces, namely those theorems that depend essentially on the fact that the product of analytic functions is again an analytic function. While there is no direct analogy for minimal surfaces, the subharmonic character of log lSI permits one to extend the proofs. We just mention two examples. Let the minimal surface S be given by the equations (3.l3) where x, y, z from a triple of conjugate harmonic functions. Suppose that 1. x(O, 0) = 0, y(O, 0) = 0, z(O, 0) = 0, 2. (x 2 + y2 + Z2)!

= lSI

:5

1 in u 2 + v2 < 1.

Then

and the sign of equality holds if and only if the surface is a simply covered circular disc. This generalizes the lemma of SCHWARZ. 111.28. Suppose this time that the minimal surface is given by v u-

S : x = x(u, v), y = y(u, v), z = z(u, v), 0 < arctg -< a, where x, y, z form a triple of conjugate harmonic functions. Suppose that these functions remain continuous for v = 0, u > 0, and suppose that x(u, 0), y(u, 0),

SURVEY OF MINIMAL SURFACE THEORY

z(u, 0) approach definite finite limits

o ::5

arctg ~ ::5 a u

E, E

141

xO, y;, z; as u - 7 + O. Then, in every angle

> 0, the functions x(u, v), y(u, v), z(u, v) approach the

limits x;, y;, z; as (u, v) -+ (0, 0). This generalizes the well-known theorem of LINDELOF. The proof follows by an extension of the so-called multiplication method!'. 111.29. The theorem ofll1.26 can be completed as follows. Threefunctions x(u, v), y (u, v), z(u, v) continuous in a domain D, form there a triple of conjugate harmonic functions if and only if log [(x + a) 2 + (y + b 2) + (z + c) 2] IS subharmonic for every choice of the constants a, b, c.

Jesse Douglas Minimal Surfaces of Higher Topological Structure Chapter 1 GEOMETRIC BACKGROUND AND STATEMENT OF RESULTS § 1. Introduction 1

About two years ago, the author published a solution of the Plateau problem for minimal surfaces of general topological form: k contours, each with an assigned sense of description, given in n-dimensional space; prescribed genus h or characteristic! r, and assigned character or orientability (two-sided or onesided).!· The publication was in the form of two papers, the first ofwhich 2 stated the results and outlined the methods used, while the second [2] gave details and proofs. Subsequently, an alternative method was outlined by R. Courant [11], who has so far published the details for the case of genus zero [12]. 3 Some years ago, the author disposed of the important special cases ofa doublyconnected minimal surface with two given boundaries [3], and of a Mobius strip I'See for instance POLYA·SZEGO: Aufgaben und Lehrsatze Vol. I p. 138 problem 277. I See the paragraph following (2.1) for definition. 10 This general form of the Plateau problem was first explicity formulated by the author in Bull. Amer. Math. Soc., v. 36 (1930), p. 50, where also is broadly indicated the method of solution here finally elaborated. 2 (I] of the bibliography at the end of this paper, references to which will be made by numbers in square brackets. 3 An independent presentation of the same method, as far as the one-contour case is concerned was given by L. Tonelli [13]. According to Tonelli, this method coincides substantially with that used by the author throughout his work on the Plateau problem. The chief point of difference consists in the use of the vector Dirichlet functional in its original form (5.12), without transformation into A(g, R).

142

THE PLATEAU PROBLEM: PART ONE

[4]. The most fundamental case of a simply-connected minimal surface bounded by a given arbitrary Jordan curve had been previously solved by the author [5], and by an alternative method, with rather less generally, by T. Rad6 [14]. Although the title of the paper [3] specified "two contours," the formulas of that paper were developed for the general case. 4 The treatment of the general problem given in [1], [2], was based on these formulas and on the theory of abelian integrals and 8-functions on a Riemann surface of any finite genus. The present paper gives, first, a self-contained treatment of the problem, reviewing and considerably amplifying the essential features of the papers [1, 2, 3]. Further, it provides certain simple supplementary considerations,4a needed only in the case of characteristic greater than one, which serve to complete the presentation of the paper [2]. This is the case where "alter-symmetric" circuits 5 are present on the basic Riemann surface R, and the Green function of R acquires certain simple complementary terms. The effect of these is easily traced, 6 with the same final result of the existence of the minimal surface. Many of the notions and formulas concerning Riemann surfaces, which are developed incidentally in the course of the analytic treatment, are new. In our main theorem, 11, 7 the existence of the minimal surface Mr is established solely on the basis of the functional A(g, R) introduced by the author-closely related to the Dirichlet functional-without any reference to area or the use of the theory of conformal mapping. In theorem I, the least area property of Mr is then proved by employing the conformal mapping of polyhedral surfaces. In § 19 an alternative method of proving the combined theorems I and II is given, which uses this conformal mapping from the start. These theorems suppose the given contours capable of bounding some surface of finite area. Finally, § 17 contains a proof of a previously announced theorem, 8 expressing the solution of the Plateau problem for the case where the contours are perfectly general Jordan curves. Solely to fix the ideas, we may suppose that the given contours do not intersect one another, but our theory applies with practically no change to the case where mutual intersections are permitted. 9 The conformal mapping of multiply-connected plane regions, which, as throughout the author's work, is included in the Plateau problem as the case n = 2, is discussed in §18. The classic mapping theorem of Schottky is reestablished by the author's method, and combined with a proof of a topological correspondence between the boundaries. This new form of treatment of the Schottky theorem was stated by the author in [l]l0 and proved in [2]. §18 of the 4 (3), p. 324. 4a(Added in proof.) These have already been published in the preliminary note (31) to the present paper. 5 See §6.1. 6 See §12. 7 See §5. 8 [2], p. 108, Theorem IV. 9 Cf. §3.3, also [8]. 10 As Theorem II of that paper.

143

SURVEY OF MINIMAL SURFACE THEORY

present paper is concerned with the details of proof of a certain essential inequality, (18.5). 2 The O-functions of the Riemann surface R enter through the expression of the Green function of R in terms of these functions. In a future paper, I intend to give a simpler and more generally applicable treatment, which will use the Green function in an intrinsic way without employing for it any explicit formula. lOa

3 For the definition of a minimal surface, we adopt the formulas given for n = 3 by Weierstrass: (i = 1, 2, ... , n)

(1.1) n

(1.2)

(w = u

+ iv).

These express only the first variational condition in the problem of the surface of least area with given boundaries, so that a minimal surface in this sense mayor may not have the least area for its boundaries. The minimal surfaces whose existence is established in the present paper will actually have minimum area for their boundaries and topological form. Thus, analytically expressed, our problem is to find a Riemann surface R of appropriate topological type, and n uniform harmonic functions .1t Fj(w) on it that obey (l.2) and represent a surface bounded by preassigned contours. IfE, F, G denote the fundamental quantities of the harmonic surface (1.1), then n

.1:

(1.3)

1

= I

F:2(w) = (E - G) -2iF. 1

Hence condition (1.2) is equivalent to (1.4)

E

= G,

F = 0,

expressing conformal representation of the harmonic surface on the Riemann surface R. It is to be observed that only .1t Fj(w) is required to be uniform; in general, Fj(w) itself will have p = r + k - 1 pure imaginary periods corresponding to various circuits on the Riemann surface. These are interpreted as translation lOa

(Added in proof.) The essential features of this alternative treatment have already been published in [32]. A detailed presentation will soon appear in the American Journal of Mathematics. See also [33].

144

THE PLATEAU PROBLEM: PART ONE

periods of the adjoint minimal surface xi =:/{F,(w). One may consider the relation between the catenoid and its adjoint surface, the helicoid, where p = and there is a single translation period. 4

If the required minimal surface Mr be considered as having two faces, corresponding to opposite senses of the normal, then it becomes a closed surface, for the two surfaces are united at the boundaries. Regarded in this way, Mr can be represented on a closed Riemann surface R, whose genus, equal to that of the double-faced M r, is seen to the p = r + k - 1. When the number k of contours is arbitrary, then even in case r = 0, the genus p of R is capable of taking arbitrary values. For this reason, nearly all the essential features of the theory for surfaces of general topological form are already present in the case r = 0, i.e., where the topological type is that of a sphere with k perforations. The rather secondary nature of the modifications that result from a higher topological structure, as will appear in the sequel, constitutes perhaps one of the advantages of the present method.

5 Illustrations of minimal surfaces of higher topological type are given in the following diagrams. All are bounded by single contours. In fig. BOa, the minimal surface has the form of a torus from which a rather large simply-connected region has been removed. II Fig BOb shows a one-sided minimal surface of the form of a Mobius strip. Fig. BOc represents a standard form of surface of higher topological type; 12 here the genus h = 2. The contour is supposed to be almost in one plane. By increasing the number of double-bridges such as AIBp A2B2, we can produce a minimal surface of arbitrary genus. Fig. BOe is a standard representation l2 of the one-sided surface; here r = 3. The half-twisted strip S produces one-sidedness and contributes unity to the topological characteristic. More contours can be introduced arbitrarily by means of additional simple bridges, or by perforations in the surface. All these examples can be realized by the soap film experiments of Plateau. The same contours also bound simply-connected minimal surfaces. Thus in BOa, we have a minimal surface consisting approximately of two horizontal circular discs joined by a vertical catenoidal strip.13 There is a second simplyconnected minimal surface, where the circular discs are vertical and the catenoidal strip horizontal. Fig BOd indicates a simply-connected minimal surface bounded by the contour of fig. BOc. This consists, very approximately, of the circular disc + the heel-shaped regions PIBIQp P 2B2Q2 + the strips AI' 11

12

13

Cf. [2], p. 122. We first gave this example in our invited address to the American Mathematical Society, Oct. 1932, published as [6] (where, however, fig. 80a is not included). [20], p.153. Cf. fig. 91 b, with the narrow connecting part removed.

145

SURVEY OF MINIMAL SURFACE THEORY

a

b

c

d

e

Figure 80

+ the shaded heel-shaped regions HI' H 2, which are to be imagined slightly above the plane of the paper. We can also visualize a minimal surface of genus h = 1 with the same contour by performing the modifications indicated in fig. BOd on only one of the double-bridges. Supposing HI> H 2, we have, very approximately, in the notation of(5.1): A2

=

(1.5)

a(r, 1)

(1.6)

a(r, 2) = a(r, 1) - 2H 2,

a(r, 0) - 2HI

which illustrate the sufficient conditions of Theorem I ( 5): (1.7)

a(r, 2) < a(r, 1) < a(r, 0),

that assure the existence of the minimal surfaces described. §2. Representation of Surfaces of General Topological form 1

This section is based on the fundamental and highly suggestive tract ofF. Klein, listed as [16] in the bibliography.14 14

Throughout this paper, the text-books [16-21] of the bibliography will be considered as current references.

146

THE PLATEAU PROBLEM: PART ONE

Let R denote a Riemann surface having an inverse conformal transformation T into itself, where T associates the points ofR in pairs, so that T2 = 1 (involutory transformation). Such a Riemann surface, following Klein, is called symmetric. The locus of points fixed under T consists of one or several closed curves, called curves of transition. If R is considered as the Riemann surface of an algebraic curve P(x, y) = 0, with real coefficients, then T may be taken to be the interchange of conjugate complex points, and the curves of transition are the real branches of the algebraic curve. The transition curves may separate R or not; following Klein, R is accordingly said to be of the first kind or second kind. If we regard a pair of T -equivalent or conjugate points as a single geometric element, then R becomes a geometric manifold R' called a semi-Riemann surface, and R' is two-sided (orientable) or one-sided (non-orient able), according as R is of the first or second kind. The transition curves of R are the boundaries of R'. Often, when there seems to be little chance of ambiguity, we shall speak interchangeably of R and its semisurface R', but quite as often it will be important to distinguish carefully between the twO. 14a Any orientable or non-orientable surface S may be put into correspondence with a symmetric Riemann surface R of the first or second kind respectively, so that a point p of S together with its antipodal p correspond to a pair of conjugate points of R. (The antipodal point of p means the point I> geometrically coincident with p, but regarded as lying on the opposite side of S, i.e., with reversed sense of the normal.) The distinction between the first and second kind of Riemann surface corresponds to the fact that on a two-sided surface we cannot pass from any point to its antipodal without crossing the boundary, whereas it is characteristic of one-sided surface that such a passage is possible. In the form of the general symmetric Riemann surface R, we are provided with a complete system of standard domains of any finite genus or topological characteristic, with any finite number of boundaries, and either orient able or non-orientable. The completeness is from the standpoint of conformal mapping, that is, the 3p - 3 real conformal moduli (p = genus ofR) are capable of varying arbitrarily in the system. From our point of view, conformally equivalent Riemann surfaces are identical, and one may replace the other if convenient for any purpose. The conformal equivalence must respect the symmetry of the surface, that is, convert conjugate points into conjugate points. What is tantamount, the corresponding real algebraic curve P(x, y) = 0 is subjct to a real birational transformation, without change of anything essential to our discussion. 14.

Remark on notation: Thus, following our papers [1, 2), we use R in this paper to represent any complete, or closed, symmetric Riemann surface of finite genus, while R' denotes the corresponding semi-surface. This convention is followed as a rule with a few occasional exceptions, that need cause no real ambiguity. In subsequent papers [31-33), we have denoted the complete symmetric Riemann surface by an Old English or script 11. and the related semi-surface by R.

SURVEY OF MINIMAL SURFACE THEORY

147

We choose R so that its semi-surface R', which results by identification of conjugate points, has exactly the topological form prescribed for the required minimal surface M, i.e., characteristic rand k boundaries, and also agrees with M in character of orientability. Then the genus of the complete surface R is (2.1)

p=r+k-l.

The definition of the topological characteristic r is: the maximum number of linearly independent circuits, where a zero circuit is considered as one which separates the surface. For two-sided surfaces, r = 2h is always even; for onesided surfaces, r may be any positive integer odd or even. IS The characteristic is also equal to the excess of the connectivity over the number of boundaries: c = r + k. Connectivity c means that there exist c - 1 successive cross-cuts which leave the surface connected, whereas every succession of c cross-cuts disconnects. A cross-cut is an arc each of whose end-points lies on a boundary or previously drawn cross-cut, while all the other points of the arc are interior points of the surface. We shall indicate the one- or two-sided character of our surfaces and quantities pertaining to them by the presence or absence of a bar. Thus we shall speak of surfaces Mr or Mr of the topological type r or F. Hereafter, we shall use r or F most frequently instead of the genus h, which applies only to orientable surfaces. Thus we may speak of the problem (r, r) or (r, F), putting in evidence the set (r) of given contours and the topological type of the required minimal surface. The following figure illustrates the case of a two-sided surface M, or symmetric Riemann surface R of the first kind, with k = 4, r = 4, therefore by (2.1), p = 7. Here the transition curves C = C I + C 2 + C 3 + C 4 separate R into two conjugate halves, and we may take R' to be the upper half. The inverse conformal transformation T may be pictured as the reflection in the plane containing C. 2 The one-sided case may be illustrated by the Riemann surface indicated in fig. 82. This is represented in the standard way as a double-sheeted surface over the z-plane. There are two branch-cuts, one C along the real axis, the other qq joining two conjugate complex points. We take the inverse conformal transformation T to be reflection in the real axis combined with passage to the other sheet. The points p. Ii in the figure represent T-equivalent points, if we take p in the upper, Ii in the lower sheet. It is evident that T leaves invariant every point of the branch-cut C, and no other points. C is therefore the unique

15

Examples: Mobius strip, r = I; with h handles, r = 2h + I; Klein surface (see footnote 25), r = 2; with h handles, r = 2h + 2.

148

THE PLATEAU PROBLEM: PART ONE

Figure 81

g··7 Figure 82

l

'la_

~I

'I.

'1.

'Is

Figure 83 curve of transition; it is to be regarded as a closed curve, proceeding from a to b in one sheet and back to a in the other. Obviously, C does not separate R, for we can pass from p to p via the branchcut qq without traversing C, as indicated in the figure. If we identify all conjugate point-pairs pp, then R represents a Mobius strip bounded by C. Point-pairs such as pp correspond to a point on the Mobius strip together with its antipodal, and the one-sided nature of the strip is expressed exactly by the possibility of passing from p to p without crossing the boundary C. By adding more branch-cuts like qq joining conjugate complex points, and branch-cuts like ab along the real axis, we can obtain models for one-sided surfaces with any characteristic r and any finite number k of boundaries. The following figure represents such a surface with two boundaries and characteristic

SURVEY OF MINIMAL SURFACE THEORY

149

three, where only the branch-cuts are indicated, along which the two sheets of the Riemann surface are joined. More generally, we may illustrate both the orientable and non-orientable cases with Riemann surfaces of any number of sheets, which admit an inverse conformal transformation T into themselves, consisting of the conjugate transformation: z into :l , combined with a substitution S of period two (S2 = 1) on the various sheets. 3 An alternative representation of one-sided surfaces Mr is based on the use of a two-sided covering surface, in the form of a semi-Riemann surface 1\' of the first kind16 with an inverse conformal transformation U into itelf, where U associates the points of R' in pairs and leaves no point of 1\' fIxed. Each pair of Uequivalent points of R' will correspond to the same point of Mr. This one-two correspondence between Mr and R' implies that the characteristic of1\' is 2r - 2 and that R' has 2k boundaries formed ofk pairs ofU-equivalent curves Cl' q; C 2, q; ... ;Ck, Ci,. The genus of the complete Riemann surface 1\ of which 1\' is one of the conjugate halves is (2.2)

p = 2r + 2k - 3.

In any parametric representation of the k contours r 1' ... ' r k on the transition curves C, each point of r j corresponds to a pair ofU-equivalent points on C j' C; respectively. For instance, a Mobius strip may be pictured in this way as a zone on the unit sphere x2 + y2 + Z2 = 1 bounded by the planes z = ± h, where diametral points of the sphere are U-equivalent.J7 Generally, we may picture 1\ as a Riemann surface of any number of sheets spread over the complex Z-sphere and consisting of two conjugate halves 1\',1\" separated by the 2k transition curves CI' q; ... ;Ck, Ci,. The inverse conformal transformation U of period two may be pictured as a diametral transformation z' = - liz combined with a substitution of period two on the sheets which compose 1\. The transformation U converts each half 1\', 1\" or 1\ into itself; besides, 1\ has an inverse conformal involution T which interchanges its two halves. T may be pictured as the reflection in the prime meridian plane, z' = z, combined with a substitution of period two on the various sheets of 1\. The transformations U and T together generate a four-group I, U, T, TU = UT of direct and inverse conformal transformations of 1\ into itself. 1\ may be regarded as a covering surface in two-one correspondence with the 16 17

That is, of the general type of the upper half of fig. 81. Cf. [4), p. 734.

THE PLATEAU PROBLEM: PART ONE

150

previously used surface 1\ (art. 2, this section), where each point of 1\ corresponds to a pair ofT-equivalent points of1\, and V-equivalent points of 1\ correspond to conjugate points of 1\. By means of the preceding representation, the Plateau problem (r, r) is referred to (r, - r, 2r - 2), where - r denotes r with the reverse sense of description; that is, we have 2k contours consisting of both orientations of each of the k given ones. We seek a semi-Riemann surface 1\' with a Vautomorphism, 2k boundaries and characteristic 2r - 2, and upon R' a system of harmonic funcions.1U'i(w), which take equal values at U-equivalent points, obey n

. 1: FP(w) = 0, 1

= I

and

transform

the

boundaries

of 1\'

in

pairs

into

the opposite orientations of each contour r. Then xi =.'R.F.1.w) or M 2r _ 2' will coincide with the desired one-sided surface Mr of type F bounded by (r). The two-side~surface M 2r _ 2 is, in fact, a covering surface in two-one correspondence with Mr' The simplest illustration is the case of the Mobius strip (r 1'1), which we solved in the paper [4] by making it depend on the previously solved case [3] ofa doubly-connected surface with two given contours: (r I' - r I' 0). Here the two contours were the two opposite orientations of the single boundary assigned for the Mobius strip. The procedure there illustrated is completely typical of the general case. By these remarks, we are enabled, as far as the analytic treatment is concerned, to confine ourselves to the two-sided case, with merely due regard to preserving the V-automorphism of 1\ when the Riemann surface is varied.

4.

Reduction of R If the symmetric Riemann surface 1\ varies continuously, it may degenerate in various ways by the coalescence of branch = points, whereby certain branch-cuts disappear. IS In this way the semi-surface R' may separate into a number of disconnected parts, or its characteristic r may be decreased. In case of separation, the total characteristic of the components is evidently::;; r. We shall term either type of degeneration a reduction of 1\ or 1\'. Every reduction can be obtained by composition of primary reductions. A primary reduction of 1\ consists of the following. (1) In the two-sided case, the disappearance of a single pair of mutually symmetric branch-cuts. (a) If the disappearing branch-cuts are the only ones that unite two given sheets, the R' separates into two disconnected parts: R' = R; + R;, with no decrease of the total characteristic: r = r l + r 2. The k transition curves C, or boundaries ofR', are distributed between R; and R;: k = kl + k 2. We may have 18

Alternatively, two branch-cuts may units to form a single one, with the same effect.

SURVEY OF MINIMAL SURFACE THEORY

151

k2 = 0, in which case R~ is a closed surface, and, as will be obvious, may simply be ignored for our purposes. If R~ is closed, we shall not call the reduction primary unless r2 = 2; ifr2 = 0, it may be considered that no reduction at all has taken place. (b) Ifbranch-cuts remain which join the same sheets as the disappearing ones, then the characteristic of R' is decreased by exactly two, or r - 2. (2) In the one-sided case a primary reduction will mean: (a) the disappearance of a single pair of mutually symmetric branch-cuts, provided that this produces separation of R':R' = R~ + R;. Here again r = r 1 + r 2. Then at least one of the components must be one-sided and both may be. (b) the disappearance of a single self-symmetric branch-cut, i.e., one which joins a pair of conjugate branch-points. This reduces the characteristic by one, to r - 1. If r - I is even, the reduced surface may be two-sided or one-sided; if r - I is odd, it must be one-sided. We shall denote primary reduction by an accent: r', r, and the general type of reduction by r", r', interpreting the latter so as to include the former. In a composite reduction of a one-sided surface, there may be any distribution of one or two-sided components. In terms of the corresponding algebraic equation P(x, y) = 0, the separation of R is equivalent to reducibility in the usual algebraic sense: (2.3) The lowering of the genus of R corresponds to the acquisition of new double points produced by the coalescence of branch-points.

5.

R' as a circular region

A useful alternative representation of a semi-Riemann surface R' is as the fundamental region of a group L of linear fractional transformations of z or i;. The k boundaries ofR' correspond to circles C p C 2, ... , C k. If the genus h = 0, R' is simply the region bounded by these k circles, and L is generated by the inversions in them: Sp S2' ... , Sk. IfR' is two-sided and h>O, we must adjoin to L a set of h linear fractional transformations of z. This gives in the fundamental region h pairs of circles Kp Kj; ... , K h, Kit, whose points are coordinated by the corresponding linear fractional transformations. If R' is one-sided, of characteristic r = 2h + I or 2h + 2, then we adjoin to Sp S2' ... , Sk a set of h linear fractional transformations of z and, respectively, one or two linear fractional transformations T of z such that T2 = 1. The latter contribute to the fuindamental region their fixed circles, whose points are coordinated by T. The complete symmetric Riemann surface R is obtained by adjoining to R' its inverse image in anyone of the bounding circles C p C 2, ... , C k, which form then the curves of transition.

152

THE PLATEAU PROBLEM: PART ONE

For instance, a torus may be represented by a circular ring, on whose bounding circles radial points are coordinated. The group L is here generated by z' = qz, if the circles are about the origin with radii I, g. Any number of boundaries on the torus may be represented by circles interior to the ring; we adjoin to L the inversions in these circles. A Mobius strip is represented by a circular ring, where diametral points of one of the bounding circles are coordinated. Equivalent to this is a zone on a sphere, where one of the bounding circles is a great circle on which diametral points are coordinated. Adjoining to this the diametral zone, we have a representation of the two-sided covering surface 1\ previously spoken of, on which V-equivalent points are diametral points and correspond to a single point of the Mobius strip. This circular representation is quite simple and concrete, but we may observe the following advantages of the many-sheeted Riemann surface. The first consists in the fact that any separation of the Riemann surface is represented directly in a visible way, while in the circular region this separation corresponds to an approach to the same point of a number of circles. Thus in the latter case, only one of the components remains visible, the other disappears; in the former, both are on a par. This circumstance is important in defining certain basic functionals depending on R in the case when R reduces. Again, in the Riemann surface representation, we have the choice of an extremely wide variety of conformally equivalent forms of a given surface, for the fundamental group is then the total real birational group, with an infinite number of parameters. In the circular representation, we have only the sixparameter group of transformations z' = (az + b)/(cz + d). We can employ this greater freedom in the former case to avoid disadvantageous types of singularity, such as reduction to a point of a curve of transition, or real branch. For the group of real birational transformations is sufficiently large and effective to enable us to neutralize any such tendency by applying, roughly speaking, a kind of balancing magnification. In the circular representation any such attempt reduces some of the previously non-degenerating circles to points (sometimes at infinity). For instance, in the two-contour case, we may use for R' the circular ring bounded by C 1: Izl = I, C 2: Izl = q < 1. There are two forms of degeneration of R', namely q = 0, q = 1. In the former case, the circle C 2 disappears, in the latter the ring itself vanishes. In contrast, we have the representation of R' as a semi-Riemann surface of a real algebraic curve of genus one with two real branches. A system of such Riemann surfaces, complete from the standpoint of conformal and symmetric equivalence, is represented, for instance, by the family of cubic curves: (x 2 + y2 - l)(x - 2) =m, (2.4)

153

SURVEY OF MINIMAL SURFACE THEORY

In this range of values of m, the curve has no double point; therefore, being cubic, it is of genus one. There are always two real branches, on which the two given contours may be represented parametrically. The modulus m corresponds to the ratio of the radii q in the circular representation. At one extreme, m = 0, the curve is reducible, separating into a circle and a straight line, each of genus zero with a single real branch. This corresponds to q = 0. At the other extreme, m = mo' the two real branches touch and the curve acquires a real double point. This corresponds to q = 1. We note that in both singular cases, the Riemann surface remains completely tangible in all respects. The figure shows the extreme curves m = m = mo drawn full, and an intermediate one dotted. We may observe that in the representation as a many-sheeted surface over the complex x-plane, one of the real branch-cuts or transition curves disappears, shrinking to the point x = 2, and the same happens to the corresponding component of the reducible surface. But if we pass to the representation over the y-plane-as we may do with preservation of conformality and symmetry-then there are two actual simplyconnected components: one with branch-points at y = ± 1, the other a simple yplane, and the second transition curve becomes the entire real axis of this plane. This illustrates how easily undesirable types of degeneration may be avoided by conformal or birational transformation of the many-sheeted Riemann surface, where this is not possible for the circular representation. The presence of infinite real branches can be avoided, if desired, by means of an inversion.

°,

Figure 84

19

Cf. [18], pp. 410-444; also [19]. The essential feature of this normal form is that all branch-points are of first order, like that of.Jz.

THE PLATEAU PROBLEM: PART ONE

154

Finally, if two of the contours r i rj have a point in common, we can represent this in the circular form by two tangent circles CiCj. But, obviously, we cannot in this way represent more than two contours with the same point in common. On the other hand, in the representation by means of a Riemann surface or algebraic curve, we can arrange for real multiple points of any order of multiplicity. For all these reasons, we shall employ in the sequel the representation by means of a Riemann surface. We may even restrict ourselves to the Clebsch-Liiroth normal form. '9 This is apart from a few occasions where the use of circles seems more convenient, but in every case the situation can be recast in terms of Riemann surfaces. Besides the many-sheeted form of R, we may consider the two-dimensional manifold S2 which is the locus in real four-dimensional space x" x2' y" Y2 of the equation P(x, + iX2' y, + iY2) = o. incidentally S2 is itself exactly a minimal surface in the four-space. Its othogonal projection on the x = x, + iX2 plane or the y = y, + iY2 plane is a Riemann surface over that plane, on which S2 is represented conformally by the projection.

§3. 1.

Orientation of the Given Contours The 2k -

1

different cases

The contours (r) must be given not only by themselves, but also each with a definite sense or orientation; in fact, a given set of k contours (r) gives rise to 2k - , distinct forms of the Plateau problem, corresponding to that number of essentially different orientations of the set. Consider first the two-sided case of a minimal surface Mr. Suppose the boundaries (C) = (C" C 2, ... , C k) of R' are oriented so that R' is on the left. Then we may orient each of the contours r" r 2' ... , r k in a definite way, and in representing (r) parametrically allow only those topological representations of each r on the corresponding C which preserve the sense, as preassigned. This is equivalent to demanding that after choice of a definite sense of the normal of Mr (which is possible, because of its two-sidedness), the contours r" ... , r k in their preassigned orientations, shall have Mr on their left, from the viewpoint of one standing on the side of the surface indicated by the directed normal, and facing in the sense assigned to each contour. It is evident that if we reverse the direction of the normal to M r , and also that of all the contours r" ..., r k simultaneously, we still have a minimal surface of topological type r bounded by the given contours, and lying on their left. Therefore, it makes no essential change in the problem to reverse the sense of all the given contours simultaneously; this has only the effect of reversing the sense of the normal on the required minimal surface, and on all the surfaces whose areas are admitted to comparison. However, if we change the sense on some but not all of the contours r" ... , r k'

SURVEY OF MINIMAL SURFACE THEORY

155

then, whichever direction we choose for the normal to M r , this surface no longer lies to the left of all the contours r I' ... , r k; i.e., it no longer furnishes a solution of the Plateau problem for the changed orientation of the given set of contours. It is evident that we may keep flxed the sense on one of the contours r I' but there are then exactly 2k - 1 different forms of the Plateau problem co:mected with the given set (r), associated with that many possible combinations of sense on the other k - 1 contours r 2 ... , r k' These 2k - 1 problems are really quite different; for instance, a given s~t of contours (r) may bound a proper minimal surface of a prescribed topological type in one system of orientations but not in another. The simplest illustration is k = 2, r = O. Here we have two given contours r I' r 2' and two essentially different Plateau problems, according to the sense which we assign to r 2 after that of r 1 has been flxed arbitrarily. Take, for instance, two co-axial circles in planes whose distance apart is sufficiently small as compared with the radii. Then if the circles are sensed as in flg. 85a, the minimal surface which is bounded by them, on their left exists, properly, being the catenoid with normal directed as in the flgure. If, on the contrary, the circles are sensed as in flg. 85b, then no proper doubly-connected minimal surface exists which they bound on their left; the only possible solution is the pair of circular discs with normal directed upward.

a Figure 85

If, for simplicity, we admit to comparison only surfaces of revolution, the difference between the two problems may be expressed as follows. In flg. 86a, the points P, Q are given on the same side of the X-axis; we require a curve joining these points, which, when revolved about the X-axis will generate a minimum area. The solution is the catenary PQ. In flg. 86b, we have the same problem with the points P, Q' on opposite sides of the X-axis, the point P being in the same position as before, where Q' is the reflection ofQ in the X-axis. Here the solution is the broken line P ABQ', as can easily be proved by elementary methods, with ise of the observation that the length of any curve joining P and Q' is greater than PA + Q'B. It is evident that if we revolve about xx' any curve

156

THE PLATEAU PROBLEM: PART ONE p

p

IJ

X'

B

A

x

x'

x

A

fJ..' b

a Figure 86

joining PQ or PQ' respectively, we obtain in either case a surface of revolution bounded by the same two circles, with respective centers A, B and radii PA, QB = Q'B. The situation may also be expressed, quite concretely, as follows. Let us take a flexible and extensible tube, as in fig. 87a, whose ends are fixed to, say, two metal rings r I' r 2 clamped in position in space. In this way a surface ~ is formed bounded by r I' r 2 and on their left, with the orientations of r I' r 2 and the normal to ~ as in the figure. Now release the connection of the tube to r 2' keeping it fixed to r I' and then stretch and bend the tube as in fig. 87b, bringing it down from above to be again attached to r 2' We obtain a surface ~' bounded on the left by the same contours r I' r 2 but in a different orientation: + r I' - r 2' Now we may put to ourselves two entirely distinct problems, namely: (a) of all surfaces derivable from ~ by continuous deformation, 20 the boundaries r I' r 2 remaining fixed, to find a minimal surface M of least area; (b) the same for all surfaces derivable similarly from ~'. In the process of deformation, we may also allow the boundaries of ~ to vary, provided that they return to their original positions induding sense. This enables us to turn a surface "inside out," e.g., to pass from ~ to ~" (fig. 87c). The normal of~" is directed towards the interior of the tube. If r I' r 2 are the two circles considered in fig. 85, then the solution of problem (a) is the catenoid, while that of problem (b) degenerates into the sum of the two circular discs. Quite as often, however, we obtain two distinct proper solutions to the problems (a), (b). Consider, for example, two circles in perpendicular planes, with coincident centers and unequal radii. Form the single contour consisting of the line-segment AB, the upper semi-circle of r 2' the line segment CD, the forward semi-circle of r l' This contour bounds a minimal surface of least area, Mi', lying entirely in the first quadrant, as we may call it. Let this surface be rotated through 180 0 about the diameter AD; then we get a minimal surface Mi', lying in the third quadrant, bounded by AB, CD and the complementary semi20

Allowing any type of self-intersection in the process.

SURVEY OF MINIMAL SURFACE THEORY

b

a

c Figure 87

Figure 88

157

158

THE PLATEAU PROBLEM: PART ONE

circles of r l' r 2' According to the Schwarz symmetry principle for minimal surfaces, 21 M~ is the analytic prolongation of of M~ across AB and CD. Therefore Mj + M~ = MI is a single analytic doubly-connected minimal surface, bounded by the circles r l' r 2' If these circles are oriented as indicated by the arrows marked MI in the figure, th~n the surface MI' with its normal properly directed, is on the left. Let MI then be reflected in the plane of the vertical circle. We obtain a minimal surface M 2, also bounded by r l' r 2' and while the sense r I is preserved, the sense of r 2 must be reversed as indicate ~ by the arrow mark-d M 2, in order to keep on thl left the surface M2 directed as obtained by the reflection from MI' If the vertical circle be moved to the right till it interlaces with the horizontal one, the preceding construction gives two (inversely congruent) doublyconnected minimal surfaces bounded by these interlacing contours. This illustrates the general theorem given by the author in [3], p. 351, according to which any two interlacing contours bound a doubly-connected minimal surface. In the present example, contrary to what one might be inclined to guess, neither of the minimal surfaces is self-intersecting. The reader will find it easy to illustrate the 22 = 4 cases corresponding to the different orientations of three given contours. 2.

One-sided surfaces Consider next the case of one-sided surfaces. The figure illustrates the two ways in which we may interpret the problem of a surface of least area with characteristic one bounded by two given contours r l' r 2 (Mobius surface with two boundaries). Here there is a proper solution only in the case of fig. 89a; in fig. 89b the solution degenerates to a Mobius strip r I and a circular disc r 2' However, if r 2' supposed to be a circle is placed with its center on the Mobius strip determined by r l' then, as I have proved, there exist proper one-sided solutions of characteristic one in both cases (a), (b). Here there can be no question of ascribing a sense to each contour so that the

a

Figure 89 21

[28], p. 175; [29], p. 397. Actually, a stronger form ~fthis sym~etry pri~c!ple is necessary for the present purpose; see J. Douglas, The analytIC prolongation of a mInimal surface over the rectilinear segment of its boundary, Duke Mathematical Journal, Dec., 1938.

159

SURVEY OF MINIMAL SURFACE THEORY

surface M is on the left, when the normal to M is properly directed, since there is no way of distinguishing over the entire surface M one direction of the normal from the other; indeed, this fact constitutes exactly the definition of a one-sided or non-orientable surface. The two senses of the normal are interchangeable when followed continuously over a properly arranged circuit. We may say here that each contour occurs with both orientations. In the representation by means of the covering surface R' with k pairs ofU-equivalent boundaries C p Cj; ... ; C k, Cit we have to decide which orientation ofrj shall go with C j and which with This gives again 2k - 1 essentially distinct possibilities, since the first choice is indifferent.

cr.

3.

Contours with common points For definiteness, we shall suppose that the given contours (r) are k Jordan curves which do not interest one another. However, our theory and results are easily adapted to the ease where the contours have points in common, provided no separate contours can be formed by combining arcs of the given ones. That is, we may have a situation like fig. 90a but not like fig. 90b. All we need do in this case is to require the symmetric Riemann surface R to have the system of its transition curves (C) topologically equivalent to the system (r) of given contours, that is, for example, in fig. 90a, the corresponding algebraic curve P (x, y) = 0 must have exactly two real branches, one with a triple point and a double point corresponding to Q3 and Q2 respectively, the other with a double point corresponding to Q2' We have already shown in detail how this works out in the case of two contours with a single point in common, and where the minimal surface is required to be simply-connected. 22 The corresponding algebraic curve might be taken to be,

a

b Figure 90

22

[8]

160

THE PLATEAU PROBLEM: PART ONE

e.g., a lemniscate, to whose real double point the common point Q of the two contours always corresponds. Actually, we used as semi-Riemann surface a parallel strip of the z-plane, whose infinite point corresponded to the intersection point Q.23 An alternative method of regarding fig. 90a is to consider that we have two contours r 1 + r 2 + r 3 + r 4 and r 5 + r 6' which, however, are not Jordan curves, but have multiple points. These can be taken account of by means of the sufficient condition (5.11) and (5.28), with the same final result as to the existence of the required minimal surface. In short, we require the semi-Riemann surface R' to be topologically equivalent to the required minimal surface Mr in all respects, including transition curves (C) as compared with given contours (r); we may say that R' is of type (r, r). If then, the k curves r l' r 2' ••• , r k which compose (r) have no multiple points either individually or mutually, then (C), as referred to the corresponding algebraic curve P (x, y) = 0, shall consist of k real branches without multiple points. On the other hand, in case the system of contours (r) presents multiple points, these must be represented as points of equal multiplicity on the system of real branches (C). §4. Surface of Assigned Topological Type Bounded by Given Contours 1.

Intuitive considerations Let the surface Sr bounded by the contours (r) vary continuously in an arbitrary way, while the boundaries remain fixed. Then, in the limit, this surface may resolve itself into separate parts, or its topological characteristic may be reduced, for example, by the closing up of an opening or by a handle of the surface shrinking to a curve. Also a closed part of Sr may simply break off, giving a closed component without relation to the contours (r). These processes may occur several times successively or in any combination, so that we may obtain most generally as limit of a surface Sr bounded by (r) a surface S" consisting of m parts having a total characteristic ~ r. If either m > 1 or the characteristic is really < r, we say that S" is the r~uced type r" and write it S; Similarly, a one-sided surface Sr boul!..ded by (r) may reduce, in varying continuously with (r) fixed, to a surface S" r consisting of one or several parts, whic!! may be one-sided or !wo-sided, having a total characteristic ~ r. We say that S" r is reduced, or type r", if it is really reduced, i.e., consists of more than one part or has a total characteristic less than r. 2.

More exact definitions Suppose given a set ofk contours (r), each with a definite sense of description. We define a surface Sr (or Sr) of characteristic r (or t) bounded 23

Cf. x 10.2.

SURVEY OF MINIMAL SURFACE THEORY

161

by (r) as follows. Let R be any symmetric Riemann surface whose semi-surface R' has the characteristic r (or t) and k boundaries (C). First, as stated in §3.3, these must be topologically equivalent to (r); for definiteness, let no two contours r and no two transition curves C intersect. On R then define any onevalued continuous vector function r (u, v) which takes equal values at conjugate points of R, and which transforms the transition curves (C) in a monotonic continuous way into (r). This means that the correspondence established by r between (C) and (r) is either one-one continuous, or deviates from this, at most, by allowing any number 24 of partial arcs of one or more transition curves to correspond to single points on the related contours. The reference is to the depiction of the correspondence by a monotonic graph as in fig. 21 (in the original text-tr.). If the correspondence is one-one in any particular case, this will be especially proved. Then the locus of the point r (u, v) in the n-dimensional space will be called a surface of characteristic r (or t bounded by (r). This surface Sr (or Sr) is two-sided or one-sided according as R is of the first or second kind in the Klein sense. The stipulation as to the coincidence of values of r (u, v) at symmetric points makes r a one-valued function on the semi-sur...face R', and we may regard r as transforming R' topologically into Sr (or Sr)' This definition gives Sr (or Sr) in a particular representation. From this, its most general representation may be derived by a topological transformation of R into itself or a homeomorphic Riemann surface with preservation of conjugate points; in other words, the semi-surface R' undergoes a topological transformation. If R is reduced, of type r' or r" , then in case of separation of R, the continuity of ~ (u, v) applies, of course, only to each component part. We have then defined a surface of the reduced type r' or r" -i.e., primary reduced or general reduced-bounded by (r). Similarly we have in the one-sided case reduced surfaces of type F', F" bounded by (r). 3.

Reduction of surfaces The simplest types of reduction are illustrated in the next figure. Fig. 91a illustrates a doubly-connected surface bounded by r I' r 2 about to reduce to the

a Figure 91 24

Finite or denumerably infinite.

b

162

THE PLATEAU PROBLEM: PART ONE

sum of two separate simply-connected ones. Fig. 91 b shows a surface of torus type about to reduce into a simply-connected one. In fig. 91c, we start with the Riemann surface for .JZ and delimit a branched portion by means of the contour r. Then we replace the branched part near the origin, bounded by 'Y, with a Mobius surface bounded by 'Y, as indicated. This gives a Mobius surface bounded by r. If now 'Y shrinks to the branch-point, this Mobius surface reduces back to the original simply-connnected branched surface.

Figure 92 A more general type of reduction is illustrated in the next figure, which represents schematically the reduction of a surface S6 bounded by five given contours into three separate parts with reduction of the total characteristic to two. One of the three handles, which gave the original characteristic six, has shrunk to a curve; the opening which produced another has shrunk to the point P; the third handle remains. The connections of the different parts have shrunk to curves, which may simply be removed, as well as the withered handle. We may also imagine additional components in the form of closed surfaces attaching at isolated points to those pictured, and these closed components may simply be broken off, without producing any new boundaries.

4.

Reduction of one-sided surfaces Starting with either a two-sided or a _ one-sided surface Sr of Sr' we may convert it into a one-sided surface Sr + 2 with an increase of two in the characteristic, by attaching a l.!.andle joining opposite sides of Sr' 25 or simply any handle in the:... case of Sr' already one-sided. Conversely, if this handle degenerates, Sr + 2 goes back to Sr or f with a reduction of two in the characteristic. Often, however, it is desirl!Ple to reEuce by exactly one unit the characteristic of a one-sided surface: _ Sr to Sr _ I or Sr _ I' and to do this by continuous deformation of Sr'

25

Fig. 93 illustrates a Klein surface with a single boundary. A Klein surface is one obtained from a sphere by attaching a handle, one end of which is joined to the outside, the other to the inside of the spherical surface. A closed Klein surface is necessarily self-intersecting, in three dimensions. See [21].

SURVEY OF MINIMAL SURFACE THEORY

163

Figure 93

Figure 94

We first consider, conversely, how to build Sr _ 1 or Sr _ 1 up to Sr' This can be accomplished by the constf!:!ction of a cross-cap, 26 as follows. We pinch up a small portion of Sr _ 1 of Sr _ 1 into the form of a cone. Then we split this cone part way down along two elements, and join the opposite edges of the two slits cross-wise in the manner of a Riemann surface. The figure indicates the successive stages of the process. It is evident that by traversing the branch-cut, we can pass from any point on the surface to its antipodal. Thus the surface has become one-sided, if it were not already so, and the characteristic has been increased by unity, corresponding to the addition of exactly one new circuit, leading from any point to its antipodal via the branch-cut. Conversely, if the cross-cap disappears by continuous variation-the split reducing to a point-the characteristic is reduced by unity, and the surface may be converted from one-sided to two-sided. This will happen if the one-sided nature is due solely to the cross-cap under consideration, and not also to topological phenomena elsewhere on the surface, i.e., other cross-caps or reverse handles. Fig. 95 illustrates, by means of cross-caps, a Klein surface with two given boundaries. The next figure shows all the forms of primary reduction. In order, these are (a) simply-connected surface r 1 + Klein surface r 2' (b) Klein surface r 1 + simply-connected surface r 2' (c) Mobius surface r 1 + Mobius surface r 2' 26

Kerekjart6 (20), Kreuzhaube.

164

THE PLATEAU PROBLEM: PART ONE

Figure 95

a

c

b

d

Figure 96

- ....'

.~.~ ." ~.>.'. /'.~:':~'~".'~>~" ~\ ~ ....

....•. -:""',,::- .-:<.""':~'" ~ .. . , . - " , . . . '.. "

.

". '

:i.:.:.,,·f

Figure 97

(d) Mobius surface I\ r 2. Fig. 97 illustrates two equivalent forms of a surface of type 3 with a single boundary. The surface being already rendered one-sided by the cross-cap, there is no distinction between a direct or reverse handle. By disappearance of the cross-cap, we obtain from (a) a torus, from (b) a Klein surface, with the given boundary. These represent the two possible types of primary reduction.

§5.

Sufficient Conditions for the Solvability of the Plateau Problem

1. The Junctional a(I', r) Consider all surfaces Sr which are properly of a given topological type r, and are bounded by a given set of oriented contours (r). Each of these has a definite area, which we understand to be defined in the sense oj Lebesgue,27 namely, as the least limit which can be approached by the area, in the elementary sense, of a variable 27

(27), Lebesgue dealt explicitly only with simply-connected surfaces. Cf. formula (16.1).

165

SURVEY OF MINIMAL SURFACE THEORY

polyhedral surface II having the topological form ofS r and which tends to Sr. We then define:

(5.1)

a(r,r) = min U (Sr)' [all Sr bounded by (r)].

Throughout, we shall use "min" as synonymous with lower bound, without prejudice of the question as to whether the minimum is attained or not. For any particular surface, U (Sr) may be finite or infinite, and, as we shall see, there are even cases where U (Sr) is identically infinite for given boundaries (r).27a Therefore a(r, r) may be finite or + 00. Consider now all reduced surfaces S; of type r" bounded by (r), whether the reduction be primary or composite. Their areas have a lower bound

(5.2)

a(r, r") = min U (S;) [all S;bounded by (r)].

We may consider als0 only the primary reduced surfaces define accordingly (5.3)

a (r, r') = min U

(S~)

[all

S~

S~

bounded by (r), and

bounded by (r)].

The previous definitions apply also to one-sided surfaces:

r, i', i" . We have (5.4)

a (r, r)

~a

(r, r").

For, given any reduced surface S", bounded by (r), it is evident that by adjunction of appropriate connecting parts in the form of narrow tubes, of direct or reverse handles, and of cross-caps, all having arbitrarily small area, we can build up S; to a proper surface Sr with the boundaries (r). From this consideration, (5.4) results immediately. We may also express (5.4) in the form

(5.5)

a (r, r) = a (r, r, r");

i.e., the lower bound of areas for the proper type r is also the lower bound for all of type r or r", proper or reduced. Evidently, in a similar manner to the passage from S; to Sr' we can build up any composite reduced surface into one of primary type, with arbitrarily small modification of area; therefore

27.

See §17.5.

166

THE PLATEAU PROBLEM: PART ONE

(5.7)

a (r, r') =s; a (r, r").

But since the set [S;] is part of the set and therefore (5.8)

[S~

], we have also the reverse inequality,

a (r, r') = a(r, r").

This is the reason why, for all essential purposes, we may consider among reduced surfaces only the primary reduced type. With (5.8), we have by (5.4), (5.9)

a (r, r) =s; a (r, r')

Throughout, all definitions and theorems apply, with the appropriate change of wording and notation, also to one-sided surfaces: T, T', T",. i.e., we may interpret r as symbolizing: either r or f. By reference to §2.4, it appears that a(r, r') is equal to the least of the following system of quantities:

(5.10) where represent all possible partitions of the set (r) and of r. In the one-sided case, a(r, r') is equal to the least of (5.10)

a(r, r=I),

a(r, r - 1),

where only even values of the characteristic apply to any two-sided component. 2. Theorem I

Suppose that for given data (r, r) we really have in (5.9), (5.11)

a(r,r)

< a(r,r').

Then we shall prove the following theorem. 27b THEOREM I. If a (r, r) is finite, and a (I', r) < a (r, r' ), then a minimal surface Mr exists which is properly of topological type r and bounded by (r). The area 27b

First stated and proved in [I], [2].

SURVEY OF MINIMAL SURFACE THEORY

167

of Mr is a minimum in comparison with all topologically equivalent surfaces Sr having the same boundaries (r), including sense. That is, U (Mr) = a (f', r). 3. The vector Dirichlet integral D(r)

Although the interest and the concrete nature of the Plateau problem certainly depend on the interpretation as a least area problem, we have found that from the standpoint of analytic treatment, it is in many ways more simple and effective to use the Dirichlet functional formed for a vector r: (5.12)

D(r) =

1

f fR' -2 (E

1

+ G) dudv = 2 =

f f (r~

+ r;) dudv

~2 f f . i= I [(aX )2 + (aXavi )2JdUdV. au r

1

In fact, it may be said that the introduction of this functional to replace the area functional constitutes the main feature of the author's contribution in the solution of the Plateau problem. 27c To define D(r) in the one-sided case, the integration is performed over the entire Riemann surface R, and the effect of this neutralized by an extra factor

1

"2'

We suppose r (u, v) to be continuously differentiable, at least piece-wise. We shall make a distinction between the area U (r) in the sense of Lebesgue, and the area as defined by an integral: (5.13)

I(r) =

f fR,v' EG

- p2 dudv.

The fact is that for surfaces r of the stated regularity, the two are equal. 28 However, in the interest of a self-contained treatment, it is sufficient to know that certainly (5.14)

U(r)~I(r),

which is much more easily-proved - see (16.11) and the associated discussion. We observe that U(r) and I(r) depend only on the surface itself, being invariant under arbitrary parameter transformation, whereas D(r) is a functional of a surface in a given representation. The parameter transformation group under which D(r) is invariant is the conformal group, as is well-known. 27c 28

cr. c. Caratheodory, Bericht aber die Verleihung der Fieldsmedai//en, Comptes Rendus du Congres International des Mathematiciens, Oslo, 1936, v. I, pp. 308-314. See [15), pp. 11-12.

168

THE PLATEAU PROBLEM: PART ONE

Let us form the harmonic vector function H(u, v) on R' with the same boundary values as r (u, v) (Dirichlet problem). Then we have by the classic minimum property of harmonic functions in the Dirichlet integral, (5.15)

D(H) :5 D(r).

Evidently, then, the problem (5.16)

D(r) = min.,

in the set [all vectors r (u, v) representing surfaces Sr bounded by (r)], is entirely equivalent to the problem (5.17)

D(H) = min.,

where r is restricted to be a harmonic vector H(u, v). The harmonic function H(u, v) depends only on the Riemann surface R and on the boundary values g(z) of H on C, which form an arbitrary parametric representation of (r). Hence D(H) is a functional of g and R, for which we use the notation. n

(5.18) i

E z

1

oR. 2 + aN.

_,

_1

AU

AU

2

dudv.

Therefore, equivalent to (5.16) and (5.17) is the problem (5.19)

A(g, R) = min.

in the set [(g, R)] consistmg of all symmetric Riemann surfaces R whose semisurfaces are of type (r, r), combined with all parametric representations g of (r) on the transition curves (C) of R. Obviously, too, (5.20)

min D (r) = min D(H) = min A(g, R) = d(r, r),

where the last symbol is the notation we now adopt for the common minimum value of these three aspects of the vector Dirichlet integral. Reduced surfaces. Evidently, D(r) may be defined just as well for the surface r (u, v) of the reduced type r', r", primary or general, as for the proper type r. We simply form, by definition, the sum of the integrals D(r) taken over the separate components of R' .

SURVEY OF MINIMAL SURFACE THEORY

169

H(u, v) with the boundary values g(z), will be determined by solving a separate Dirichlet problem for each component part of R'. Accordingly, if (5.21) then, by definition, (5.22) where gj denotes the part of g that belongs to the transition curves of R j. This is the reason that any closed components Rj may be ignored, since a function harmonic and everywhere regular on a closed Riemann surface must reduce to a constant and therefore contributes nothing to the Dirichlet integral. We now define d(r, r'), d(r, r") as the common minimum value of D(r) or D(H) = A(g, R), when the Riemann surface R varies over all of the reduced type r', r" respectively, and g varies arbitrarily. Evidently, since type r" includes r', we certainly have d(r, r') ~ d(r, r"), but the fact is, as will be proved, that d (r, r') = d (r, r").

(5.23)

which, as stated before in connection with area, justifies confining ourselves to the primary reduced type. Dirichlet functional and area functional. We observe the identity: (5.24)

.!. (E + G) 2

-J EG - F =

.!. (El 2

- Gl)2

+ .!.1-JEl Gl + F - -JE! G! - F12.

2 This proves immediately: THEOREM (a). D(r)

~

I(r). If D (r) = I(r), then E

=G, F =0, and conversely.

In other words, D(r) = I(r) is the necessary and sufficient condition for a conformal representation. By the first part of the theorem and (5.14), we certainly have (5.25)

d(r, r)

~

a(r, r).

d(r, r')

~

a(r, r').

Also, for reduced surfaces, (5.25')

The fact is, however, as will be proved, that

THE PLATEAU PROBLEM: PART ONE

170 (5.26)

d(r, r) = a(r, r);

also d(r, r') = a(r, r').

(5.26')

4. Theorem II We shall also easily prove the analogue of (5.9): (5.27)

d(r, r) :s; d(r, r').

Suppose that for given data (r, r), we really have the strong form of the inequality: (5.28)

d(r, r)

<

d(r, r').

This constitutes the sufficient condition in the statement of our main theorem. 28• THEOREM II. If d(r, r) is finite, and d(r, r) < d(r, r'), then a surface Mr bounded by (r) exists, in a representation Mr (u, v) on a semi-Riemann surface properly of topological type r, such that D(Mr (u, v)) = d(r, r). Every such surface is minimal, and its representation is conformal. The spirit of our method consists in proving this theorem exclusively on the basis of the functional A(g, R), without any reference to area, or the use of any part of the general theory of conformal mapping.

5. Arbitrary Jordan contours If the contours (r) are arbitrary Jordan curves, then in general (5.29)

a(r, r)

=+

00,

I.e. In order that a(r, r) be finite, a necessary and sufficient condition is that each contour rj be capable of bounding a simply-connected surface of finite area: (5.30)

283

See footnote 27b.

a(rj) = a(rj' 0)

< + 00,

j = 1, 2, ... , k.

SURVEY OF MINIMAL SURFACE THEORY

171

For this in turn a sufficient condition is, evidently, that each rj be rectifiable; then, for example, the cone obtained by joining all the points of rj to any fixed point of space has finite area. To prove the equivalence of a(r, r) < + 00, and (5.30), we observe first that (5.31) since the right member represents a particular type of reduction of Sr' namely, into k simply-connected parts. Therefore (5.30) implies that a(r, r) is finite. Conversely, let a(r, r) be finite; then a surface Sr exists bounded by (r) such that U (Sr) is finite. We have then only to assume that for each contour r j, it is possible to construct on Sr a rectifiable closed curve r; topologically equivalent to rj by continuous variation on Sr. This is certainly the case if the defining function r (u, v) of Sr is continuously differentiable piece-wise. Then the ringshaped portion of Sr between rj and q, combined with any conical surface having for base, evidently gives a simply-connected surface of finite area bounded by r j. Conversely stated, the necessary and sufficient condition for a(r, r) = + 00 is that a(rj) = + 00 for at least one of the contours r j. For the case where a(r, r) is finite, we define the functional

q

(5.32)

e(r, r) = a(r, r') - a(r, r)

~

°

by (5.9). For arbitrary Jordan curves r, whether capable of bounding surfaces of finite area or not, we define (5.33)

e (r,

r) = max lim e(r m' r)

~

0,

the maximum being with respect to all possible ways of approaching (r) with a sequence of contour systems (r)m' m = 1, 2, 3, ... , such that each contour of each system is capable of bounding a finite area. This mode of approach is always possible, since the approaching contours may be taken to be, for instance, polygons. Now, the following theorem holds. 28b THEOREM III. Suppose a(r, r) = + 00. Ife (r, r) > 0, then a minimal surface Mr exists which is properly of topological type r and bounded by (r). The area of Mr is infinite, but every completely interior 29 portion M' has an area which is finite and a minimum for its own topological structure and boundaries. 28b 29

Announced in2, p. 108. Meaning interior inclusive of its boundaries.

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THE PLATEAU PROBLEM: PART ONE

This theorem will be proved in § 17. In other words, our sufficient condition is: (1) for the case a(r, r) finite, that a(r, r') - a(l', r) be strictly positive; (2) for the case a(l', r) = + 00, that the limit of a(r, r') - a(r, r) be strictly positive, for some way of tending to the given contour system (I') with a sequence of contour systems (I')m such that each approaching contour is finitearea-bounding. Throughout this discussion we may replace the area functional by the Dirichlet functional and a(l', r) by d(l', r).

III SOME FACTS FROM ELEMENTARY TOPOLOGY §1. Singular and Cellular Homology Groups 1. Singular chains and homology groups. The results obtained in recent years in connection with the solution of topological variational problems have shown the importance of homology groups. We shall therefore briefly recall some necessary facts about these groups, and give algebraic and topological constructions to be used in the sequel. Consider the Euclidean space Rk + I with Cartesian coordinates XP- •• ,xk + l' and also the standard simplex .lk of dimension k, given thus: xI + ... + xk + I = 1; xI ~ O, ... ,xk + I ~ 0. DEFINITION I. A continuous mapping f of the standard simplex .lk into a topological space X is called a singular simplex fk of dimension k of the space X. A formal linear combination c = fagof singular simplexes fk of a space X with integral coefficients, of which only a finite number are different from zero, is called an integral k-dimensional singular chain c of the space X. The set of all kdimensional chains clearly forms an additive Abelian group C k (X) called the group of k-dimensional chains of the space X. A homomorphism defined on the generators of the group thus: (\f k = i ~ 0 k (- 1YfJ' - I, where fJ' - I is the restriction of the mapping fk to the i-th face of the simplex .lk, is called the boundary operator a k: C k (X)~ C k _ I (X).

It is clear that a k a k + I == 0, therefore Ker a k :J Ima k + I. The factor group Ker ak/lmak + I is called the k-th singular homology group Hk (X) of the space X. The elements of the subgroup Ker a k are called cycles, and those of the subgroup 1m a k + I boundaries. DEFINITION 2.

It easily follows from the definition of singular homology groups that they are topologically and homotopy invariant, i.e., are unaltered under a homeomorphism of the space or on replacing it by a homotopy equivalent space. The collection of groups Ck (X) and the homomorphisms ak is connecting them is naturally organized into the sequence ... -+Ck ~Ck _ 1-+ ..........C I .i4 Co 4Z -+0, where ak ak + I == 0, and E is an epimorphism. This sequence is called a chain complex. A mapping of one chain complexinto another, commuting with the homomorphisms a k, is called a chain mapping. We say that a chain homotopy D ofa complex C into a complex C', connecting two mappings q> and \V of the complex C into C' is given (where

THE PLATEAU PROBLEM: PART ONE

174

if a collection of homomorphisms D k : C k Ck+ I is given so that, for any k, we have: Dk _ I dk + dk+ I Dk = Q>k - \Ilk. Two chain mappings Q> and \II connected by a chain homotopy are sometimes said to be chain homotopic. It follows from the definition of homology groups that chain homotopic mappings induce the same mappings of homology groups Hk (C) into Hk (C'), where Hk (C) = Ker dk/lmd k + I· To compute homology, the so-called exact sequence of the pair is useful. Let topological spaces X, Y be given, where Y is a closed subspace of X. It is clear that C k (Y) C C k (X), and we can consider the group of relative chains Ck (X)/C k (Y) = C k (X, Y). Since the boundary operator acts as follows:

it induces an operator C k (X,Y)--+Ck _ I (X,Y) which, for simplicity, we denote by the same symbol dk. The groups Ker dk = Zk (X, Y) ::J Bk (X, Y) = 1m dk (relative cycles and relative boundaries) can be defined, which makes it possible to consider the factor group Hk (X, Y) = Zk (X, Y)/Bk (X, Y)

called the group of relative k-dimensional homology of the space X modulo the subspace Y. It can be easily verified that, for relative homology groups, topological and homotopy invariance again hold. We now pass to the construction of the new operator d: Hk (X,Y)~Hk _ I (Y). Let zk €Ck (X,Y) be a relative cycle, and Z k € C k (X) an arbitrary representative of it in the coset. Since ok Zk = 0, dk Z k € C k _ I (Y). We denote this absolute cycle by dZk€ Zk - I (Y). The homology class of this cycle does not depend on the choice of representative from the homology class of the cycle zk. We can therefore define in this way a homomorphism (operator) d: Hk (X,Y)--+Hk _ I (Y) which we shall also denote by d, and call a boundary (homomorphium operator Fig. 98). Further, denote the embedding by i: Y --+ X. Then it induces a homomorphism i.: Hk (Y)~ Hk (X). Since any absolute cycle can be considered to be relative (modulo the subspace Y), this gives rise to the natural mapping:

THEOREM I. The following sequence of groups and homomorphisms is exact, i.e., the image of an antecedent homomorphism coincides with the kernel of the subsequent homomorphism at each of its terms: Hk

+ I

(X,Y) 0 Hk (Y) i* Hk (X) j* Hk (X,Y) 0 Hk -

I

(Y)

Proof. Let us verify the exactness, e.g., at the term Hk (X,Y). If 001 = 0,01 € Hk

FACTS FROM ELEMENTARY TOPOLOGY

175

x

Figure 98

(X, Y), then oZk is homologous to zero in Y (where Zk is the representative of ex, i.e., Zk € Zk (X, Y)). But then, by adding to zk a k-dimensional chain that realizes this homology in the subspace Y, we obtain a k-dimensional chain which is a cycle already from the point of view of the space X, i.e., we have represented the element ex as the image of an element {3, i.e. ex = i. {3, {3 € Hk (X) (Fig. 99). Thus, Ker 0 C Imi •. The reverse inclusion follows from the fact that any absolute cycle can be considered as a relative one in X, having zero boundary in Y. The exactness of the sequence for other terms is verified by a similar argument. For the sequel, it is useful to know the following properties of exact sequences (we leave the proofs to the reader): (1) The sequence O..... A..... O is exact ifand only

Figure 99

THE PLATEAU PROBLEM: PART ONE

176

if A = O. (2) The sequence O--.,A~B--.,O is exact if and only if the groups A and B are isomorphic, and the homomorphism a is an isomorphism. (3) The sequence 0- A.J.B.lItC_O is exact if and only if the group A is a subgroup of the group B, the homomorphism i: A ~ B being an inclusion (monomorphism), C = BfA (factor group), and 11": B~BfA the natural projection onto the factor group. It turns out that relative homology can be reduced to absolute homology.

2. Cell complexes, barycentric subdivisions. IfX,Y are path-connected, then Ho (X,Y) = o. A topological space X is called a cell complex if it is representable as the union of disjoint subsets ok called cells, the closure of each cell ok being the image of a closed k-dimensional disc Dk under a certain continuous mapping (called characteristic) which is a homeomorphism onto the interior of the disc. Further, the boundary of each cell (i.e. the image of the boundary of the disc under the characteristic mapping) must be contained in the union of a finite number of cells of lesser dimensions, i.e., not exceeding k-l. Finally, it is required that a subset in X should be closed if and only if all full inverse images (under the characteristic mappings) of the intersections of this subset with all the cells are closed. A cell complex is said to be finite if it consists of a finite number of cells. The union of cells of dimensions not exceeding k is called the k-dimensional skeleton of the complex. We will say that a pair of spaces (X,Y) is a cell pair if X and Yare cell complexes, and the closed subspace Y is a cell subcomplex in X. We confine ourselves to considering finite complexes. Denote the space obtained from X by identifying the closed subspace Y with one point by X/Y. THEOREM

2.

Let (X, Y) be a cell pair. Then Hk (X,Y) = Hk (X/y) when k

:1=

O.

Proof. Denote the cone over Y by CY, i.e., space obtained from the cylinder Y x I by contracting the upper base to one point (Fig. 100). Construct a new space X U CY, i.e., identify the subspace Y in X with the base of Y in the cone CY (Fig. 101). Since X,Y are finite complexes, by contracting the cone CY on itself to a point, we obtain a homotopy equivalence: X U CY "'" XIY (Fig. 102). Thus, to prove the theorem, it suffices to establish that, for k > 0, the relations hold: Hk(X, Y) = Hk (X U CY). It follows from the exact sequence of the pair (X U CY, *), where * is a point, viz., the vertex of the cone, that Hk (H U CY) = Hk (X U CY,*) for k > 0; therefore, we have to prove that

for k > O. Before the proof, we shall need an argument relating to barycentric subdivisions.

FACTS FROM ELEMENTARY TOPOLOGY

177

-

·..·:~:2:··~r:?}~~z(~::-~· y Figure lOO

Figure 101

Figure lO2 Let Ak be the standard simplex. Its barycentric subdivision (3 Ak is defined by induction as follows. If k = 1, then (3 A I is obtained by adding a new vertex in the middle of the line segment AI, Ifk = 2, then (3 A2 is obtained by joining the centre of the triangle to its vertices and the mid-points of its sides (Fig. 103). And, finally, for an arbitrary k, the subdivision (3 Ak is obtained as follows: mark

178

THE PLATEAU PROBLEM: PART ONE

pAi O------~O~-------O

Figure lO3 the centre of .:;lk, and break the simplex .:;lk into pyramids with the vertex at this centre and the bases-simplexes of the barycentric subdivision on the boundary of the simplex .:;lk. Now let f: .:;lk~X be an arbitrary singular simplex of the space X. We denote the chain equal to the sum of all singular simplexes obtained by restricting the original mapping f to the k-dimensional simplexes of the barycentric subdivision f3.:;lk of the original simplex .:;lk by f3f. Associating each singular simplex f (Le., elementary chain in C k (X)) with the singular chain f3f, we obtain the homomorphism 13k: C k (X)-+C k (X). We claim that the collection of mappings 13 = {13k} defines a chain mapping 13 of the chain complex C(X) into itself; moreover, it is chain homotopic to the identity. That the mapping 13 is chain follows from the fact that the restriction of the formal sum of singular simplexes to the barycentric subdivision equals the formal sum of these restrictions. We now construct the chain homotopy D connecting the mapping 13 with the identity explicitly. Define the division of the prism .:;lk xl into the sum of simplexes for each k > 0 as follows. This division is shown in Fig. 104 for k = 0, 1, 2. The inductive process of division of .:;lk x I can be described as follows. If the division is already defined for q < k, then it should be taken on a part of the boundary of .:;lk x I, viz. the union ("cup") (.:;lk x 0) U (a.:;lk x I) so that .:;lk x 0 may be the standard simplex, and a .:;lk x I divided in accordance with the inductive process when q < k. After that, we decide the whole prism into kdimensional simplexes whose bases are (k-l)-dimensional simplexes of the indicated division of the part of the boundary, and the vertex is the centre of the upper face. It is clear that the upper face will then undergo a barycentric subdivision. Now, let f: .:;l~X be a singular simplex. We denote by Dk f E C k + I(X) the (k + 1)-dimensional chain, that is the sum of all (k + 1)-dimensional singular simplexes obtained by restricting the mapping <1>: .:;lk x I Y, where (x, t) = f{x), to the simplexes of the above-constructed division of the cylinder .:;lk x I. We obtain the homomorphisms D k: C k (X)~Ck + I (Y) which define the required chain homotopy connecting the identity mapping of the complex C(X) onto itself with the barycentric mapping 13. The proof that the mappings {Dk} give a chain

FACTS FROM ELEMENTARY TOPOLOGY

179

Figure 104 homotopy is left to the reader. We now pass to the proof of the theorem. Consider the pair-embedding mapping (X, Y)~ (X u CY, CY). It induces the homomorphism a: Hk (X, Y)-+Hk (X U CY, CY) = Hk (X U CY, *). We have used the fact that CY .., *, i.e., the cone is contractible to a point. We next prove that a is an epimorphism. Let z EO Zk (X U CY, CY) be some cycle. We have to find in the group Zk (X, Y) a cycle which will be carried into a cycle homologous to z in Zk (X U CY, CY) under the indicated mapping. Represent the cone CY as the union of two subsets: a cone A, i.e., part of CY consisting of points, for which t consisting of points for which t

~

~

!

!,

and a truncated cone, i.e., part of CY,

(Fig. 105) Reducing the singular simplexes

making up the cycle z by means of barycentric subdivisions, we each time obtain cycles homologous to it, but consisting of ever finer and finer simplexes. Since

180

THE PLATEAU PROBLEM: PART ONE

Figure 105

the number of simplexes is finite, and X, Yare compact, there exists a sufficiently fine barycentric subdivision such that if any singular simplex of this subdivision such that if any singular simplex of this subdivision intersects A, then it wholly lies in the cone CY. Here we consider the image of the standard simplex under the mapping specifying a singular simplex. This cycle z' is homologous to z. Discard all those simplexes which intersect A from the cycle z' . Since this operation is performed within the cone CY, from the point of view of relative homology the new cycle z" remains in the same homology class as z'; hence, z also. Thus, z" E Hk (X U B, B,). At the same time, because of the homotopy invariance of homology groups, we have: Hk (X U B, B) = Hk (X, Y), since (X U B, B) "" (X, Y) (Fig. 106). Thus, we have explicitly exhibited a certain cycle z" E Hk (X, Y), which is carried into the cycle z EHk (X U CY, CY) by the homomorphism CY. The surjectivity of CY is thereby proved. The proof of monomorphicity is performed in an analogous way, and is left to the reader. See Fig. 107.

3. Cellular homology and computation of the singular homology of the sphere. We have described above one of the ways of introducting homology groups, using singular simplexes. However, for cell complexes, the so-called cellular homology can be defined, which coincides, as it happens, with singular homology, but possesses one important advantage: cellular homology is essentially easier to compute. Even in the simplest cause where the space consists of one point, the computation of singular homology requires a certain (but elementary) argument. If, however, the space is arranged in a more complicated way, then the computation of its singular homology rapidly becomes more complicated. In practice, cellular homology is used particularly often precisely for this reason.

FACTS FROM ELEMENTARY TOPOLOGY

181

Figure 106

Figure 107 We introduce it on the basis of the concept of singular homology already known to us, which will then enable us not only to prove the coincidence of singular and cellular homology groups, but also specify an explicit scheme for their computation. First, we compute the singular homology of the sphere. Since the zero-

182

THE PLATEAU PROBLEM: PART ONE

dimensional sphere consists of a pair of points, we obtain: Hk (SO) k ;:0, and Ho (SO) = ZESZ.

=

°

when

LEMMA 1. The singular homology groups of the n-dimensional spheres sn, where n > 0, are of the form: Hk (sn) = when k ;:0, n, and Hk (sn) = Z when k = 0,

°

n.

The proof follows from the consideration of the exact homology sequence of the pair (Dn + I, sn) and from the fact that a disc is contractible to a point. Recall the definition of the wedge of two topological spaces. Let two points xo and Yo' respectively, be distinguished in two spaces X and Y. Construct a new space X V Y by identifying these points (Fig. 108). No other points are identified.

y

x XvY Figure 108

LEMMA 2. Let X = ViSf be the wedge of the n-dimensional spheres Sfwith the subject i, where 1 :s; i :s; N. If n > 0, k > 0, then the isomorphism Hk (X) = i Hk (SP) = z ... Z (N times) holds. For the proof, it su.ffices to consider the exact sequence of the pair (Vi Df; Vi oDf) with Vi (DfIODf) = Vi Sf(Fig. 109). Note an important fact for the sequel. We can take as a generator in the group Z = Hn (D,sn - I), the homology class of the simplest singular chain l·f, where f: Lln Dn is a homeomorphism of the simplex on to the disc. Therefore, the orientation of the sphere can be specified by fixing a generator in the group

FACTS FROM ELEMENTARY TOPOLOGY

183

Figure 109 Z = Hn (sn). A change in orientation is equivalent to the replacement of the element 1 by the element - 1. We now define cellular chain groups. Let X be a finite cell complex. Let us try to compute its singular homology in terms of cells and their characteristic mappings, i.e., in those terms in which the complex is given. The set of all kdimensional cells of a complex X will be denoted by X k. Let Xk be the kdimensional skeleton of the complex X. We shall assume that the orientations of all the cells are fixed. Number all the k-dimensional cells, and let Ak be the set of all subscripts. Then, due to Lemma 2, we obtain: H. (X k Xk - 1) = H. (V Sk) = { 0 when i +k I , I (tEA a Pk(X) when i = k. k

Here, P k (X) denotes a free Abelian group whose generators are in one-to-one correspondence with Ak. Since the elements of this group are naturally identified aaa~, where the a~ are the k-dimensional with linear combinations of the form cells of the complex X, the group Pk(X) is finitely generated. We call this group the group of k-dimensional chains of the space X. The groups P k (X) and Ck (X) are not isomorphic in the general case. Before going further, we consider the socalled exact homology sequence of the triple, which is a variant of the sequence of the pair. Let (X, Y, Z) be three spaces, where Y and Z are closed in X. Consider the two embeddings (Y, Z)--t(X, Z) and (X, Z)--t(X, Y); let

F

a:

Hk (X, Y)---tH k _ 1 (Y, Z)

be the boundary homomorphism generated by the homomorphism

a:

Hk (X, Y)--t Hk -

1

(Y)

(whose definition is given above) and also by the fact that each absolute cycle from Hk _ 1 (Y) can be considered as relative modulo Z, i.e., as an element of the group Hk _ 1 (Y,Z). Then we obtain the sequence

THE PLATEAU PROBLEM: PART ONE

184

Hk (X, Y) a Hk _ 1 (y, Z) Hk _ 1 (X, Z) Hk _ 1 (X, Y) a ... The verification of its exactness is left to the reader. Returning to the groups P k (X) = Hk (Xk/Xk - I) and Pk _ 1 (X) = Hk - 1 (Xk - I/X k - 2), we can consider the exact sequence of the triple (Xk, Xk - I, Xk - 2). For the moment we only need from it the homomorphism Hk (Xk, Xk - I) ~Hk _ 1 (Xk - I, Xk - 2), which is written in our notation thus: P k (X)~Pk - I(XJDenoting it by iJ k, we obtain the chain complex k (X), iJkJ, viz., ...-Pk (X) .....k Pk _ 1 (X)~..... Just as for any chain complex, its homology groups are defined, i.e., the groups Ker iJ/ImiJ called the cellular homology groups of the complex. As it happens, there exists a canonical isomorphism between the homology of the chain complex described and the singular homology of X. This is the basic statement of the present item, enabling one to reduce the computation of singular homology to the computation of the homology of a considerably simpler chain complex. As it happens, this reduction is so effective that most concrete computations of homology are based just on this theorem. In particular, it immediately follows that cellular homology groups are homotopy invariant, and that singular homology groups of a finite complex are always finitely generated.

lp

4. Theorem on the coincidence of the singular and cellular homology of a finite cell complex. THEOREM 3. For a finite cell complex X, the singular homology groups Hk (X) and homology groups of the chain complex {Pk (X), iJ k} i.e., the groups KeriJk/ImiJ k + 1 (the so-called cellular homology groups) are isomorphic. First, we prove certain auxiliary statements. LEMMA 3. When k

> 1, the following isomorphism holds:

Proof. Consider the triple of complexes (Xk corresponding exact sequence:

+

I, Xk - 2, xk - 3) and the

Hk (Xk - 2, Xk - 3) Hk (Xk + I, xk - 3) Hk (Xk + I, Xk - 2) Hk _ 1 (Xk - 2, xk - 3). Its extreme terms are equal to zero, i.e.,

Repeating the argument for the triple (Xk + 1, Xk - 3, Xk - 4), we obtain

FACTS FROM ELEMENTARY TOPOLOGY

185

Proceeding further with respect to dimension, we obtain the following chain of isomorphisms: Hk (Xk + I, Xk - 2)

= Hk (Xk + I, Xk -

3)

= Hk (Xk + I, Xk - 4)

= ... = Hk (Xk + I, XO) = Hk (Xk + I)

when k > 1. If the skeleton XO consists of only one point, then the equality is valid also for k = 1. The point is that any finite cell complex (connected) is homotopy equivalent to a finite complex such that its zero-dimensional skeleton consists precisely of one point. For this, it suffices to consider all the zerodimensional cells of the original complex, and join each of them to one distinguished vertex (zero-dimensional cell *); moreover, all these paths must lie in the one-dimensional skeleton XI. Then we carry out the homotopy shown in Fig. 110, by contracting all the zero-dimensional cells to one.

Figure 110

LEMMA 4. The following isomorphism holds: Hk (X) = Hk (Xk + 1). < k + 1, the equality holds: Hi (Xk + I) =

Proof. We prove that, for any i Hi (Xk + 2).

In fact, consider the exact sequence of the pair (Xk + 2, Xk + 1):

Then the required equality follows from this; in particular, Hk (Xk + I) = Hk (Xk + 2). Going back to the proof of Lemma 3, we obtain:

which completes the proof. LEMMA 5. The following isomorphism holds: Ker 8k/lm8k + I = H (Xk + 1,

186

THE PLATEAU PROBLEM: PART ONE

Xk - 2), where Ok are homomorphisms defining the complex of cellular chains. Proof. Consider the commutative diagram:

Here, the row is a segment of the exact sequence of the triple (Xk + 1, Xk, Xk - 2), while the column is a segment of the exact sequence of the triple (Xk, Xk - 1, Xk - 2) and the homomorphisms i*, i* are induced by the corresponding embeddings of the pairs i,i. The commutativity of the diagram means that Ok + 1 = i*o. Recall that Per (X) = Her (Xer, Xer - 1). Since the row and column are fragments of exact sequences, i* is an epimorphism, and i* a monomorphism. Hence, Hk (Xk + 1, xk - 2) = Hk (Xk, Xk - 2)/Ker i* = Hk (Xk, Xk - 2)/lmo. Since i* is a monomorphism, Hk (Xk, Xk - 2)/lmo = i* Hk (Xk, Xk - 2)/i* Imo = Imi*IImi* = Ker ok/1m i* = Ker Ok/1m Ok + 1 Here, we have used the identities 1m i* = Ker Ok (because of the exactness) and i* = Ok + 1 (commutatively of the diagram). Thus, Hk (Xk + 1, Xk - 2) = Ker 0klIm Ok + " and the lemma is proved.

°

°

°

Proof of Theorem 3. We obtain from Lemmas 3 and 5 Hk (X) = Ker ok/1m Ok + 1 = Hk (Xk + 1, Xk - 2), thus completing the proof.

5. The geometric determination of cellular homology groups. We need to clarify the geometric meaning of the operator Ok in the chain complex {Pk (X), Ok}' Consider two cells Ok - 1 and Ok in X. We will assume that their orientations are fixed and that the characteristic mapping x: D k X, x: D k - 1 X are compatible with these orientations. Consider the continuous mapping. ODk = Sk -

1

~ Xk - l/Xk - 2,

i.e. we map the boundary of the cell into the factor space of the (k - 1)-dimensional skeleton with respect to the (k - 2)-dimensional skeleton.

FACTS FROM ELEMENTARY TOPOLOGY

187

This factor space is homeomorphic to the wedge of (k - 1)-dimensional spheres. Since we have distinguished the cell Ok - I in the (k - 1)-dimensional skeleton Xk - I, it is mapped onto a certain sphere Sk - I from the wedge Xk - IlXk - 2 under the indicated factorization Xk - I a Xk - IlXk - 2. Thus, one sphere is distinguished in the wedge of spheres, which makes it possible to define the natural projection of the whole wedge onto this sphere. That is to say, the distinguished sphere is not displaced, and all the others are mapped into the base point (Fig. Ill). The mapping so constructed of the boundary of the ball Dk into the wedge of spheres with subsequent projection onto the distinguished sphere defines a continuous mapping Sk - I Sk - I, and, in particular, determines a homomorphism of the group Z = Hk _ I (Sk - I) into itself. Each homomorphism Z -Z is uniquely determined by an integer m, viz., the image of the identity element of the group Z. This number is called the degree of the mapping. In the case where the mapping constructed by us is smooth (it will be so in many examples considered below), the number m coincides with the usual degree of a smooth mapping, defined for mappings of orientable closed manifolds of the same dimension [50].

Figure III

We have associated each pair of cells Ok and Ok - I with a certain integer called the incidence coefficient of the cells and usually denoted thus: [Ok: Ok-I]. It can be seen from its definition that this number depends on the orientations of the cells chosen by us, and changes sign on changing one of the orientations. THEOREM 4. Let Ok be an arbitrary generator of the group Pk (X) = Hk (Xk, Xk - I). Then the action of the boundary operator 0 on this generator is given by the formula OOk = E[Ok:Ok - 110k - I, where the sum is taken over all the (k - 1)-dimensional cells Ok - I of the complex X. This statement provides us with a clear geometric interpretation of the boundary

188

THE PLATEAU PROBLEM: PART ONE

operator a introduced above in the algebraic language for cellular chain groups. If the cell Ok - 1 does not intersect the closure of the cell Ok, then [Ok: Ok - I] = o.

Proof of Theorem 4. Consider two triples: (Dk, Sk - 1,0) and (Xk, Xk - I, Xk - 2), and the continuous mapping (Dk, Sk - 1,0) (Xk, Xk - I, Xk - 2), where Dk Xk is the characteristic mapping of the cell Ok, and Sk - 1 Xk - 1 is the restriction of this mapping to the boundary of the ball. The exact sequences of these triples can be naturally organized into the following commutative diagram:

o

z

z

I I" Hk (Dk)-+Hk (Dk,

1 i.

Sk -

. "Hk _ 1 (Sk - 1)---+0

I)~

1~.

Hk (Xk, Xk - I)~ Hk _ 1 (Xk - I, Xk - 2)

"

Pk (X)

"

Pk

- 1

(X)

Consider the element 1 EO Z = Hk (Dk, Sk - I). Under the homomorphism i., this generator is mapped into the cellular chain 1.o k EOP k (X), and after applying is transformed into 1.a Ok. Let us follow the displacement of this generator along the upper side of the square. Under the mapping j, the element 1 is transformed into 1 EO Z = Hk _ 1 (Sk - I) (since j is an isomorphism). Under the subsequent mapping ~., the generator of the group Z is transformed into a certain element of the group

a,

To each generator of the group P k _ 1 (X) = Hk _ 1 (VSk - I), there corresponds a certain cell Ok - I. It is clear that the coefficient of this cell in the image of the identity element under the mapping ~. is exactly equal to the degree of the composite mapping Sk - 1 Sk - I, i.e. the coefficient [Ok: Ok - I]. The theorem is thus proved. So, we have obtained a rather simple rule for computing the singular homology groups of a cell complex. For this, it suffices to consider the chain complex of cellular chains uniquely determined by the cell structure of X, and then write out the boundary operators explicitly, for which it suffices to compute the incidence coefficients of pairs of cells of consecutive dimensions. Then, the homology groups of the complex of groups so obtained must be computed. This construction is so vivid (and easily computed in many cases) that it sometimes forms the basis for the definition of cellular homology groups.

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189

DEFINITION 2. Let X be a finite cell complex, Pk (X) cellular chain groups, and a k: P k (X) P k - 1 (X) homomorphisms defined by the formula a k Ok = E [ak:a k - I] ok - I. Then the homology groups of this complex, i.e. the groups Ker a k + 1 are called the cellular homology groups of the complex X. To compute these groups, the singular homology of X need not be known, since all the objects involved in Definition 2 admit of a purely geometric description (cells, characteristic mappings, incidence coefficients). Theorem 3 can be reformulated thus: the singular and cellular homology groups of a finite cell complex are isomorphic, with the corollary that if one space X is represented as a cell complex in two ways, then the cellular homology groups of X do not depend on the cellular decomposition, since they are isomorphic to the singular homology of the space. Thus, to compute the homology of the space X, we must choose its representation as a cell complex in as simple a way as possible, and then compute the cellular homology.

6. The simplest examples of cellular homology group computations. EXAMPLE I. The sphere sn admits of the simplest cellular decomposition: 0° U an, where 0° is a point, and an its complement in the sphere. It is clear that, for n~ 1, we have: Hi (sn) = Z when i = 0, n, and H j (sn) = when j #=0, n.

°

EXAMPLE 2. The real projective space Rpn. Recall that one of its realizations is the set of sequences of the form x = (xo> xl''''' xn), where Xi are real numbers and at least one coordinate is non-zero, the sequence being considered up to a nonzero multiplier. The simplest cellular decomposition ofRpn is arranged thus: for the cell Ok, all sequences x should be taken, for which xk #=0, xk + 1 = ... = xn = 0. Then, in each dimension k, we obtain exactly one cell Ok, i.e., Rpn = 0° U 0 1 U ... U an. Therefore, Pk (Rpn) = Z. It remains to compute the boundary operator ak:z Z. Fig. 112, the closure of the cell Ok, i.e., Rpk, is represented as a k-dimensional ball whose boundary, viz. the sphere Sk - 1, is factorized with respect to the action of the group Z2' i.e., the generator of this group is represented by the transformation x - x, the reflection of the sphere in the origin. In other words, Rpk is obtained from the ball Dk by identifying diametrically opposite points on its boundary. Meanwhile, the boundary of the ball Dk is mapped into the factor Rpk - l/Rpk - 2 = Sk - 1 as follows. Represent Sk - 1 as the union of three disjoint subsets S~ - I, Sk - 2, Sk- 1 _, where S~ - 1 and SI: - 1 are open hemispheres (upper and lower), and Sk - 2 the equator. The mapping aDk = Sk - 1 Sk - 1 = S~ - I/Sk - 2 is arranged thus: x x ifx E S~ - I, X * when x E Sk - 2, -x x when -x E Sk- I. Thus, we have obtained a mapping h: Sk - 1 Sk - 1 = S~ - 1/Sk - 2 which is a diffeomorphism on each of the subsets S~ and S~· It remains to fmd the degree of this mapping. It is clear that

190

THE PLATEAU PROBLEM: PART ONE

the inverse image of each point x EO S + consists of two points: the point itself and the point diametrically opposite to it on the sphere Sk - 1. Therefore, the required degree either equals two or zero according to whether the orientation of the sphere Sk - 1 changes or not under the mappings x ~ - x. Therefore, it is required to find the degree of the auxiliary mapping a: Sk - 1-+ Sk - 1, where a (x) = -x. LEMMA 6. The degree of the mapping a (x) = - x of the sphere Sk equals (-I)k.

1

onto itself

Hence, it immediately follows that the degree of the mapping h equals 2 for even k, and zero for odd k. Thus, for odd k, the coefficient [Ok: Ok - 1] equals zero, and two for even k. This means that the boundary operators in the chain complex {Pk (X), «J k} have the form «J0 2-Y = 202-y - 1 = o. Thus, we have proved the following statement. Proposition I. The singular (and cellular) homology groups ofRP have the form:

Hn

o

for even n Z for odd n.

In conclusion, we prove Lemma 6. Consider the sphere Sk - 1, and fix an arbitrary tangent orthogonal frame e (x) = (ep ... ,ek _ 1) at a point x. Under the

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191

mapping a: x ~ - x, this frame will be transformed into the frame e ( - x) = (- el"'" - ek _ I) (we assume that the sphere is standardly embedded into Euclidean space). We need to compare the orientations induced on the sphere by these two frames. Join the points x and - x with a meridian 'Y such that its velocity vector at the point x is the vector ek _ 1 (Fig. 113). Carry out a smooth deformation (transfer) of the frame e(x) from the point x to the point -x by moving along the path 'Y so that the vector ek _ 1 remains tangent to 'Y. Then we obtain two frames at the point - x: - el"'" - ek _ 2' - ek _ 1 and - el"'" - ek _ 2' + ek _ l' It is clear that their mutual orientation is determined by the sign of (- 1)1t - 2, thus the proof is completed.

Figure 113 EXAMPLE 3. The two-dimensional, compact, connected, closed, and orientable manifold M~ of genus g is homeomorphic to a two-dimensional sphere with g handles, and admits of the cellular decomposition a O U (~12g at> U a 2. It follows that the homology of M~ is of the form:

(2g terms), H2 = Z. The homology groups defined above do not embrace the whole set of "homological invariants" that sometimes enable one to distinguish between cell complexes. A homology theory can be constructed with the use of chains with coefficients in an arbitrary Abelian group A. This means that chains should be considered as linear combinations with coefficients in A. It is clear that all the

192

THE PLATEAU PROBLEM: PART ONE

constructions are automatically transferred to this case, enabling one to define the groups Hk (X, A) called the homology groups with coefficients in the group A. The results relating to cell complexes are valid for the group A = Z (considered above) as well. In particular, these groups can be defined as the homology of the chain complex made up of the groups of the cellular chain groups {Eaj OJ' ajE A} = Pk (X,A) with coefficients in A. The homology groups Hk(X) studied by us above are written in the new notation as Hk (X,Z). For brevity, we will often omit the designation of the coefficient group except in those cases where the final result depends on the choice of the group A. EXERCISE. For Examples 1,2,3 (see above), calculate the homology groups with coefficients in the groups A = R, Q, Z2' Zp' where R is the field of real numbers, Q the field of rational numbers, and p a prime number.

§2. Cohomology Groups and Obstructions to the Extensions of Mappings. 1. Singular cochains and the coboundary operator. Let X be a cell complex, and C k (X) the group ofk-dimensional singular chains of the space X. Let A be an Abelian group. DEFINITION I. A homomorphism of the group C k (X) into the group A is called a singular cochain of the space X with coefficients in the group A. The natural operation of cochain addition transforms the set of cochains into an Abelian group denoted by Ck (X,A), and called the group of cochains of the space X. Consider the boundary operator 0: C k (X)-+C k _ 1 (X). Let hECk - 1 (X, A) be an arbitrary cochain, i.e., a homomorphism h: C k (X)-+A. Then the cochain 5h E Ck (X, A) is determined uniquely. It is given by the formula: 5h(a) = h (0 a), i.e., 5h: C k (X) -+A. In other words, the cochain 5h is determined from the diagram.

We will sometimes denote the operator 5: Ck -

1

(X, A)-+Ck (X, A) by 15 k _ I'

DEFINITION 2. The operator 15k _ I: Ck - I-tCk is called the coboundary operator. It is the dual of the operator o.

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FACTS FROM ELEMENTARY TOPOLOGY

Since iF = 0, 02 = O. Therefore, we obtain a sequence of groups and the homomorphism connecting them, which is of the following form:

where ok Ok _ 1 = O. This sequence is called a cochain complex. Proceeding as in the previous section, consider the groups Ker 0 and 1m 0, and construct the group Hk (X, A) = Ker 0k/lmok - l' DEFINITION 3. The groups Hk (X, A) are called the cohomology groups of the space X with coefficients in the Abelian group A. The elements of the group Bk = 1m Ok _ 1 are called coboundaries, and those of the group Zk = Ker Ok cocycles.

If the space X is path-connected, then HO (X, A) = A. As in the case of homology, the relative cohomology groups are defined in natural fashion. Let Y be a closed subcomplex of the complex X, then Ck (Y) C Ck (X). Let 0 (X, Y) be the group of all those homomorphisms a: C k (X)~A that are equal to zero on Ck (Y). It is clear that 00 (X, Y) C 0 + 1 (X, Y), and this therefore gives rise to the groups Hk (X, Y, A) = Ker o/lmo. They are called the relative cohomology groups. As in the case of cohomology, the exact sequence of the pair and triple arise. Omitting the details of construction, we only give the exact sequence of the pair (verify that it is exact!): ... -+Hk (X, Y, A)-+Hk (X, A)~Hk (Y, A)~ Hk + 1 (X, Y, A)-+... and also of the triple (X, Y, Z): ... ~Hk (X, Y, A)4Hk (X, Z, A)4Hk (Y, Z, A)~Hk

+ 1

(X, Y, A)-+ ...

Singular cohomology groups are homotopy invariant. Proceeding as in §1, we define cellular cohomology groups. To this end, we introduce the cellular cochain groups pk (X, A) defined as the groups Hk (Xk, Xk - 1, A), where Xk is the k-dimensional skeleton of the cell complex X. The exact sequence of the triple (Xk + 1, Xk, Xk - 1) gives rise to the coboundary operator 0: pk (X, A)~ pr- + 1 (X, A). Cellular cohomology groups are defined as the groups Ker O/Im for the cochain complex {Pk (X, A), o}.

°

THEOREM I. For a finite cell complex X, the singular cohomology and cellular cohomology groups are isomorphic, and are finitely generated Abelian groups if the coefficient group is Abelian.

The proof is as given in § 1.

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THE PLATEAU PROBLEM: PART ONE

2. The problem of the extension of a continuous mapping from a subspace to the whole space. While studying topological variational problems (to which the subsequent sections are devoted), we shall sometimes have to solve the following problem. Let Y be a closed subspace of a topological space X, and let a continuous mapping of this subspace into some space Z be given. The question arises: in which cases can this mapping be extended to a continuous mapping of the whole space X into the space Z? It is clear that such a continuous extension does not always exist. There are topological obstructions which, in certain cases, do not permit us to extend a mapping from a subspace to the whole space. Obstruction theory takes up the study of all the various versions of this question. Here, we only give the results which will be used in the sequel. We already possess all the necessary material to give an account of the required constructions.

3. Obstructions to the extension of mappings. Consider a finite cell complex K and a topological space X. Let a continuous mapping g of the (n - I)-dimensional skeleton Kn - 1 of the cell complex K (Le., union of all its cells of dimensions not exceeding n - 1) into the space X be given. We intend to extend it to a mapping f of the following skeleton Kn in X. For simplicity, we will assume that X is either a simply-connected space or, in case X is one-dimensional, possesses a commutative fundamental group. Consider all n-dimensional an which make up the n-dimensional skeleton of the complex K. Since it is finite, the number of these cells is finite. To construct the required extension, we must be able to extend the original mapping to each ndimensional cell separately. Fix some cell an. Its boundary has already been mapped by the mapping g into the space X. It has to be extended to the whole cell. We will assume that the mapping f: Kn~x under construction coincides with the mapping g on the skeleton Kn - 1 C Kn. Let x: on~K be the characteristic mapping of the cell an, where on is an n-dimensional disc. Then we have the composite mapping sn - 1 = a on_ X-+Kn - 1_ g-+X. Here, we use the fact that the boundary of the cell an is contained in the skeleton Kn - 1 in accordance with the definition of a cell complex. The mapping g x: sn - 1-+ X so obtained determines an element [gX] of the homotopy group 11"n _ 1 (X). This is well-defined in view of the restrictions imposed on the space X. They guarantee that the definition of the elements of the group 1I"n _ 1 (X) does not depend on the choice of a base point. In the general case, we must require that the space X be (n - I)-simple (see the definition in [84]) Le., that the fundamental group act trivially (from the homotopy point of view) on 1I"n _ 1 (X). See this action in Fig. 114. It is clear that, with the above assumptions, X is (n - I)-simple. The mapping gXn: Sn - I-+X can be extended to the mapping of the whole disc on in that and only that case where the element [gX] from 11"n _ 1 (X) equals zero. In the general case, we associated each cell an with a certain element [gx] of the Abelian

FACTS FROM ELEMENTARY TOPOLOGY

195

Figure 114

group G = 'II"n _ 1 (X). Consider the group Pn (K) of the cellular chains of the complex K. It is clear that the constructed correspondence (Jn~[gX] f G can be naturally extended to a certain homomorphism of the group of chains P n (K) into the group G. For this, it suffices to construct the mapping for each cell, and extend it to all chains by linearity. We thereby define a certain cochain c~ from the group of cellular cochains. DEFINITION 4. A cochain c~ f pn (K) with coefficients from the Abelian group G = 'II"n _ 1 (X) is called an obstruction to extending the mapping g from the skeleton Kn - 1 to the skeleton Kn. LEMMA I. A mapping g: Kn - l-+X can be extended to a continuous mapping f: Kn~x if and only if the cochain c~ is identically equal to zero. The proof follows from the definition of an obstruction. LEMMA 2. Let a space X be (n - I)-simple. The cochain c~ E pn (K, 'll"n _ 1 (X» is a cocyle, i.e., oc~ = O. Therefore, the cochain c~ defines a certain cohomology class C~ = [c~] from the group Hn (K, 'll"n _ 1 (X». Since the ideas underlying the proof will not be required by us in the sequel, we omit the proof [84]. In the applications that we shall encounter, the equality oc~ = 0 will follow from an obvious argument. THEOREM 2. Let a space X be (n - I)-simple. A cocycle C~ equals zero (as an element of the cohomology group) if and only if the original mapping g: Kn - l--?X can be extended to a continuous mapping f: Kn~x by preliminarily

196

THE PLATEAU PROBLEM: PART ONE

changing the mapping g on the skeleton Kn skeleton Kn - 2.

I,

and not changing it on the

In contrast with Lemma 2, the plan of the proof of Theorem 2 will be used in applications; hence, we give the proof. Proof. To construct the mapping f, we shall need a new notion closely related to the obstruction c~, viz. the difference cochain. Let two mappings f, g: Kn - I~X coinciding on the skeleton Kn - 2, i.e., f (x) = g (x) for all x € Kn - 2, be given. Let an - 1 C Kn - 1 be an arbitrary cell, and x: Dn - 1--+ K its characteristic mapping. Since the boundary sn - 2 of the disc Dn - 1 is mapped into the skeleton Kn - 2, the two composite mappings fX and gX map the sphere sn - 2 into the complex K in the same way. The cell an - I is mapped differently, generally speaking; however, these, two images have a common boundary, being glued on the image of the sphere sn - 2 in K. Therefore, in the space X, we obtain a spheroid which determines a certain element of the group 7rn _ 1 (X) and measures the deviation of the mapping f from the mapping g on an - I. In this way we have associated each cell an - 1 with an element of the group 7rn _ 1 (X) (here, the (n - I)-simplicity of X is used). We obtain a homomorphism of the group of chains P n _ 1 (K) into the group 7rn _ 1 (X), i.e. an (n - I)-dimensional cochain. This cochain is denoted by dp'g- I, and called the difference cochain of the two mappings f and g (Fig. 115). It follows from its definition that dp'g- 1 = 0 if and only if there exists a homotopy connecting f and g and constant on the skeleton Kn - 2 where f and g coincide.

fJ

Figure 115

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197

LEMMA 3. For any mapping f: Kn - I_X and any cochain a f pn - I (K, 1I"n _ I (X», a mapping g:Kn - I~ X can be always chosen, so that it coincides with f on the skeleton K n - 2 and the cochain d is the difference of f and g, i.e., d = dp'g- I.

Proof. Consider an arbitrary cell an - I from K and its image under the mapping f in X. Choose a sufficiently small ball in the centre of the cell, and, considering its image under f in X, cut this image (using the cellular approximation theorem [84]) out of the image of the cell an - I. Then glue to the hole so obtained a spheroid, a representative of that element from the group 11"n _ I (X), which is the value of the cochain d on the cell an - I (Fig. 116). As the new mapping g, take the mapping coinciding with f everywhere except on the distinguished ball, and coinciding on the ball with the mapping realizing the indicated spheroid. Roughly speaking, the distinguished small ball bulges into the spheroid realizing the value of the cochain d on this cell. Performing this operation on each cell, we obtain a certain mapping g. Comparing it with the original, we obviously obtain d = rlP,g= I. The lemma is thus proved.

G' n-i Figure 116

LEMMA 4. The equality holds: Odp'g-

I,

=

q -

C~.

Proof. Due to the definition of a coboundary operator in terms of cellular cochains and chains, we have the equality:

198

THE PLATEAU PROBLEM: PART ONE

To compute the incidence coefficients, we have to consider the following composite mapping

where 11" is the natural factorization, and Pi the projection on the wedge of spheres onto the sphere with number i in this wedge. The degree of the mapping sn - I-,+Sr - 1 so obtained is what is called the incidence coefficient [on: or - 1]. It is easy to prove that the mapping sn - I--+Kn - 1 is homotopic to a mapping under which almost the whole sphere sn - I, with the exception of a finite number of small balls, is mapped into the skeleton Kn - 2, and the small balls are mapped into the skeleton Kn - I, each small ball being mapped onto its cell or - 1 with degree ± 1. The algebraic number of the small balls mapped onto the cell or - 1 (i.e., sum of degrees ± 1) is exactly equal to the required incidence coefficient (Fig. 117). In the figure, the thick line segments on the boundary of the ball schematically represent the small balls of dimension n - 1, which are mapped onto the cells or - I. Let us calculate the value of the cochain cp - c~ on

Figure 117

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199

the cell on (Fig. 117), i.e. where the boundary cells or - 1 are mapped. Eventually, we have to determine where the small balls distinguished on sn - 1 are mapped under the composite mapping involving f and g. Since the mappings f and g coincide on the skeleton K n - 2, to compute the value of the cochain on n , i.e., all the the cell on, we have to take the summation over the boundary cells or - I, of the following spheroids from 11"n _ 1 (X). On each cell or - I, we have to take the spheroid coinciding with the value of the difference cochain dp'g- 1 on the cell or - I, and as many times as there are small balls mapped onto the cell or - I. Obviously, this sum gives the incidence coefficient, and the whole cell on will therefore be associated with the sum E[on: or - I] dp'g- 1 (or - I) coinciding with (0 dP,g- I) on, which completes the proof.

ao

Let us return to the proof of Theorem 2. Let the mapping g: Kn - 14 X be extended to the mapping f: Kn_x without being changed on the skeleton Kn - 2, but possibly, being changed on the skeleton Kn - I. Due to Lemma 4, we have: n - 1 = C n - cn But since f is defined on Kn cn = o· therefore cn = odf,g f g. , 'g" g Odp'g- I, so that C~ = O. Let us prove the converse statement. Suppose that C~ = 0, i.e., c~ = Od, where d is a certain cochain. According to Lemma 3, there exists a mapping f:Kn - I-+X coinciding with the mapping g on the skeleton Kn - 2, and such that -d = dp'g- I. Then, due to Lemma 4, we obtain q = C~ + Odp'g- 1 = c~ - od = o. Therefore, the mapping f can be extended to the mapping Kn~x. Thus, we have extended to the mapping g, possibly having a priori changed it on the skeleton Kn - I, but not on Kn - 2, which finishes the proof.

4. The cases of the existence of the retraction of a space onto a subspace which is homeomorphic to the sphere. We prove the retraction theorem which we shall need for the study of variational problems. Consider the Abelian group (additive) U = RI (mod 1), i.e. the circumference, as the coefficients of the cellular homology theory. THEOREM 3. (Hopf). Let an embedding i of the sphere sn - 1 into a finite ndimensional cell complex K be given. Let the homomorphism i*: Hn _ 1 U)-+Hn _ 1 (X, U) induced by this embedding be a monomorphism. Then the sphere S~ - 1 = iS n - 1 is a retract of K, i.e. there exists a continuous mapping f:K-+ S~ - 1 which is an extension of the identity on the sphere S~ - I. Proof. The constructive variant of the proof given here, necessary for the explicit construction of retractions is due to T. N. Fomenk030o • We have to construct a continuous mapping f:K~ sn - 1 such that its restriction to the sphere iS n - 1 embedded into K is the identity mapping of the sphere iS n - 1 onto sn - I. To construct such a mapping, we apply obstruction theory. First of all, choose the simplest cellular decomposition into the sum of two cells for the sphere sn - I:

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THE PLATEAU PROBLEM: PART ONE

zero-dimensional * and (n - I)-dimensional an - I. Therefore, we can construct the required mapping so that it is cellular and transforms the whole skeleton Kn - 2 into a point on the sphere sn - I. In other words, it suffices to construct a continuous mapping of the quotient complex K/Kn - 2 into the sphere sn - I; here, K = Kn. To apply obstruction theory, we have to take X to be an (n - I)-simple space. Take X = sn - I, then the condition of(n - I)-simplicity is obviously fulfilled. In fact, if n - I > I, then 7r 1 (X) = 0, and if n - I = I, then the group 7r 1 (S I) = Z is Abelian and acts on itself as the identity. Since the skeleton Kn - 2 has been contracted by us to a point, we can assume that the skeleton Kn - 1 coincides with the wedge of(n - I)-dimensional spheres s~

-

1

V sy -

1

V .... V

S~

-

I.

It is clear that the sphere iS n - 1 can be considered to be one of them. Let it be, for definiteness, the sphere S~ - I. Represent each of the spheres Sr - 1 as the simplest cellular decomposition, and denote the corresponding (n - I)-dimensional cells by or - I. We must construct a continuous mapping f:K~sn - 1 such that its restriction to S~ - 1 is the identity mapping onto sn - I. Let oy, ... , o~ be n-dimensional cells of the complex K. Then we may assume that nI 1 n K -- o-vo o

1

U... UOkn - IU0 nu' I_n 1 ••• vu q

Figure 118

(Fig. 118)

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201

Preliminarily, we must construct a certain mapping of the skeleton Kn - 1 into sn - I. Now, we construct one such mapping gl which, generally speaking, will not be extended to the mapping of the whole K into sn - I, but can be used for constructing a second and final mapping g2 extendable to the mapping f:K~ sn - I. Since the skeleton Kn - 1 is the wedge of spheres, it suffices to specify the mapping gl on each of them. We specify the identity mapping S~ - 1 ~ sn - 1 on the distinguished sphere S~ - I, and map all the remaining spheres into the base point * f sn - I. This mapping is continuous; however, it is easy to construct examples showing that it cannot be extended to a continuous mapping of the whole complex into the sphere. Let us calculate the obstruction C~I for the mapping gl· Consider an arbitrary n-dimensional cell on. To calculate c~ (on), we have to consider the composite mapping of the (n - I)-dimenslcinal sphere, the boundary of the cell, into the sphere sn - I, and find its degree. Here, we use the fact that 1I"n _ 1 (sn - I) = Z, and that the homotopy classes of a mapping of a sphere into a sphere are classified by the degree of the mapping. This composite mapping is the composite of the characteristic mapping and gl :Kn ~ sn - I. That part of the boundary sphere (Jon, which was mapped into the wedge S? - 1 V .... V S~ - I, i.e. which is not incident with the distinguished sphere S~ - I, will be mapped, finally, into the base point * on the sphere sn - I. Therefore, these portions of the sphere (Jon do not take part in forming the required degree (contribution is zero). The remaining part of the boundary sphere (Jon is mapped onto the distinguished sphere S~ - 1 "twisted" on it as many times as the incidence coefficient [on: o~ - I], and then mapped onto the sphere sn - 1 with the help of the identity mapping. Finally, the boundary sphere (Jon is mapped onto the sphere sn - 1 with the degree equal to the incidence coefficient [on: o~ - I]. Thus, we have calculated that C~I (on) = [on: o~ - I]. We state that this cochain is representable as the coboundary of a certain (n - I)-dimensional cochain d. LEMMA 5. With the assumptions of Theorem 3, the cochain C~I is a coboundary, i.e., C~I = a d, where d f P n - 1 (K,1I"n _ 1 (sn - I) and d (M~ - I) = 0, O~A~ 1. Proof For simplicity, consider the case when, among the n-dimensional cells of the complex K, there is only one cell on with non-zero incidence coefficient with the distinguished cell o~ - I. Roughly, the further argument will be developed as follows. Our purpose is to construct an (n - I)-dimensional cochain d on the wedge of spheres, so that this cochain must be equal to zero on the distinguished cell c~ - 1 (more precisely on the elementary chain 1. o~ - I), and, generally speaking, non-trivial on the remaining spheres of the wedge, i.e., we want to "remove" the cochain from the distinguished sphere S~ - I. We shall use the fact that the cycle S~ - 1 is not homologous to zero in the complex K for the coefficient group U. Recall that the homomorphism induced by an embedding of the sphere has a trivial kernel. We compute the boundary of the cell on. We have:

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THE PLATEAU PROBLEM: PART ONE

Denoting [an: ar - I] by aj. we can write:

Thus, the chain - ao a~ - I is homologous to the chain Qj a~ - I + ... + Qk a~ - I. Therefore, the cycles defined by them coincide as elements of the homology group Hn _ I (K, Z). It follows from the conditions of the theorem that the cycle }.a~ - I, where 0 <}.< 1, is non-zero in the group Hn _ I (X,D). Consider the following three cases: (a) all the integers al' ... ,ak are co prima (b) the numbers al' ... ,ak have a greatest common divisor p different from unity, the number ao/p not being an integer, i.e., p does not divide ao; (c) the numbers al' ... ,ak have a greatest common divisor different from unity, and p divides ao. Consider case (a). Since the numbers a!, ... ' ak are coprime, there exist integers x!' ... , xk such that ao = XI a l + ... + xk ak• Define an (n - 1) -dimensional cochain d on the skeleton Kn _ I thus:

Here, the numbers xl' xk are realized as the degrees of mappings of the spheres Sr - I onto the sphere sn - I. It is clear that 0 d = C~I. In fact, (Od)a n = d (<Jan) =

j

E k Q da. n - I = = I ,1

E~ I

=

I X.Q· = Q0 = engl ( n) 1 1 (J

•

Thus, the lemma is proved in case (a). In case (b), consider the equality

where aj = pbj,l :si:sk, and the numbers b p •.• , b k are coprime. Since the number ao/p is not an integer, using the coefficient group D, we find that there exists a non-zero cycle

a _0

a~

- I (when}. = lip) in the homology Hn _ I (Sr - I,

P

D), which becomes homologous to zero in the group Hn _ I (K, D), since ao 1 an - I == <J (- an) (mod 1). This contradicts the fact that pop the fact that the homomorphism Hn _ I (sn - I, D)~ Hn _ I (K, D) is monomorphic. Therefore, case (b) is not actually realized. Finally, consider case (c). Since p is the greatest common divisor of the numbers a!, ... ak, it follows that aj - pbi' l:s i:s k, ao - mp, me Z, and the numbers b p ... , bk are coprime. Therefore, there exist integers xl' ... , xk such that m = XI b l + ... + xk b k • Realize the numbers Xj as the degrees of the mappings of the spheres S~ _ I into the sphere Sf - I, and define the cochain d as follows:

FACTS FROM ELEMENTARY TOPOLOGY

d (or - I)

= Xjf1l"n _ I (sn -

I),

d(o~

- I)

203

= o.

We assert that a d = c~!, In fact, (ad) on = d (ao n ) = d (aoo~ - I + ... + ako~= alx l + ... + akxk = mp = ao = c~ (on). The lemma is thus proved. I

1)

Let us return to the proof of Theorem 3. Due to Lemma 3, for the mapping gl: Kn-I-+sn-I and cochain d constructed above, i.e., such that ad = C~I' a continuous mapping g2: Kn-I-+sn-I can be chosen so that the equality d = d n- Ig g holds. Due to Lemma 4, we have: adn - Iglg2 = ad = C~I - C~2' i.e., C~2 = 10~ Thus, we have found a mapping g2 of the (n - I)-dimensional skeleton Kn into sn-I, which is extendable to a continuous mapping f:Kn-tS n- l . It remains to see that the mapping is the identity on the sphere S~ - I. This follows from the proof of Lemma 3. In fact, the mapping g2 is constructed thus: it is different from gl only on those (n - I)-dimensional cells, on which the cochain d is nonzero. But the cochain d has been specially constructed so as to be zero on chains of the form AO~ - I; therefore, g2' as well as gl' is the identity mapping of the sphere S~ - I onto the sphere sn - I. If there are several n-dimensional cells incident with the cell o~ - I, then the proof is given in a similar manner, and we omit it, leaving it to the reader. The theorem is thus proved. Despite its geometric simplicity, the given proof is rather bulky. We have chosen this path, since it will prove useful to us in the sequel, when we need to construct a concrete retraction of a certain complex onto the sphere. The explicit construction of the required retraction becomes elementary after the above argument. However, it seems pertinent here to give another and shorter proof of Theorem 3, less effective and not so vivid, but clarifying other general topological requirements for the existence of the required retraction. We shall have to make use of the properties of the so-called Eilenberg-MacLane complexes, though. Fix a natural number n 2: 1 and group 11" assumed to be Abelian ifn> 1. Then there exists a cell complex denoted by K (11", n) and called the Eilenberg-MacLane space, such that all its homotopy groups are trivial except for the group 11" (K(1I", n», which is isomorphic to the group 11". The following important statement holds: the set n (K, K (11", n» of the homotopy classes of mappings of a finite cell complex K into the space K (11", n) is in one-toone correspondence with the set of elements of the cohomology group Hn (K, 11"). Consider the sphere sn - I, and embed it into the space K (Z, n - 1) as an (n - I)-dimensional skeleton. For this purpose, a certain number of cells should be glued to the sphere in order to "eliminate" all its homotopy groups beginning with dimension n. Meanwhile, we use the fact that 1I"n _ I (sn - I) = Z. The first homotopy group which needs annihilating is the group 1I"n (sn - I). Therefore, we have to glue cells of dimensions n + 1, n + 2, etc., to the sphere sn - I. Eliminating all higher homotopy groups one after another, we obtain a certain complex B, for which 1I"j (B) = 0 when it=n - 1 and 1I"n _ I(B) = Z. It is well

204

THE PLATEAU PROBLEM: PART ONE

known that any two polyhedra which are spaces of type K( 11", n - I) for given n - 1 and 11" are homotopy equivalent. It is clear that Hn - l(K (Z, n - 1), Z) = Z; K(Z, n - 1) = UOOUO n - IUOn + IUO n + 2U... Let a€ Hn - I(K(Z, n - I)Z) be the generator of the group Z. Then (see [84]) the function associating the mapping f:K-+K(Z, n - 1) with the homology class f'l'a € Hn - 1 (K, Z) for any finite complex K determines a one-to-one correspondence between Hn - 1 (K, Z) and the set II (K, K(1I", n» of homotopy classes of mappings of K into K (11", n). The condition of Theorem 3, according to which the homomorphism i.: Hn _ 1 (sn - I,U)-+H n _ I(K,U) has no kernel, can be restated in an equivalent manner in the language of cohomology, viz., the homomorphism i*: Hn - 1 (K, Z)+Hn - 1 (sn - I, Z) induced by the embedding is an epimorphism. Therefore, there exists an element {3€ Hn - 1 (K, Z) such that i* B = a€ Hn - 1 (sn - I,Z). Because of this, there exists a continuous mapping f:K-+K (Z, n - 1) such that f'l'a = {3. Since (fi)*a = a, the restriction of the mapping f to the sphere iS n - 1 embedded into K is homotopic to the identity mapping of the sphere iS n - 1 onto itself. It remains to show that the mapping f can be deformed into a mapping which remains the identity on the sphere sn - I, and will map the whole complex K onto it. But since the complex K is n-dimensional by assumption and since the complex K (Z, n - 1) does not contain cells of dimension n, it follows from the cellular approximation theorem (see, e.g., [84]) that the mapping f can be assumed to transform K into the (n - I)-dimensional skeleton of the space K (Z, n - 1), i.e. the sphere sn - I, which completes the proof of Theorem 3.

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217

INDEX Absolute maxima 78 Area functional 53, 172

Douglas theorem 54 Drop's geometry 14-15

Barycentric subdivision 180 Bernstein's problem 137 Bernoulli Jacob 2 Bifurcation of the minimal surface 37 Boundary conditions 36 Boundary contour (frame) 8, 9 Boundary operator (in homology theory) 173 Boyle experiments 14 Branch-points of the minimal surface 127, 166

Edge of regression 73 Eilenberg-MacLane space (complex) 203 Enneper minimal surface 36, 49 Equation of the minimal surface 66, 131 Euler-Lagrange equation 35 Euler Leonhard 2 Evolute 73 Exact sequence of the groups 174 Extremal properties of soap films 13 Extremals of the functional 35

Catalan's theorem 27 Catenary line 32, 87 Catenoid 32, 88 Cauchy-Riemann equations 137 Cell-complex 176 Cellular chains 183 Cellular cohomology 193 Cellular homology 180, 184 Chain complex 173 Characteristic of minimal surface 161 Characteristics 68, 69 Coboundary operator 192, 197 Cochain complex 193 Cohomology groups 192 Complete catenoid 90 Complete Riemannian surface 152 Conformal (or isothermal) coordinates 17,52,53 Conformal representation of the minimal surface 169 Convex boundary 51, 132 Critical points 21,36 Critical surface 25 Cross-cup 53

Figures of revolution 86 First fundamental form 16,49 Functional 35 Gaussian curvature 16 Generalized harmonic function 126 Genus of the Riemannian surface 149 Handle 52-54 Harmonic mappings 51,52 Harmonic radius vector 52 Helicoid 27 Homology boundaries 173 Homotopy groups 173 Homology cycles 173 Homotopic mappings 174 Homotopy of chain complexes 173-174 Induced metric 16 Isothermic parameters 133 Jordan curve 127 Jumps of the minimal films 38-40 Klein surface 163

Darboux theorem 136 Developable surface 73 Dirichlet integral 50, 168 Dirichlet functional 36, 51,52, 169 Dirichlet principle 53 Differential equation of the minimal surfaces 66 Double-curvature curve 73

Lagrange 3 Lagrangian 35 Laminar systems 97 Laplace equation 50 Laplace operator 17 Lines of curvature 73 Liquid edges 96 219

220

INDEX

Many-valued functions 35 Mean curvature 16 Membranes 62 Meusnier (Jean-Baptiste-MarieCharles) 6, 7 Minimal radius vector 52 Minimal two-dimensional surfaces 20 Minimal surfaces in animate nature 60 Minimal surface in three-dimensional space 50 Minimal surface of assigned topological type 160 Mobius strip 53 Mobius surface 167 Monge Gaspard 3, 65 Multiplicity of the point of the minimal surface 57 Multiplicity of the self-intersection 57 Non-parametric problem 136 Obstructions to the extention of mappings 194-195 One-sided Riemannian surface 147 One-sided minimal surface 159-160 Orientation of the minimal surface 154-155 Partial catenoid 86 Physical realization of minimal surfaces 27 Plateau Joseph 8, 83 Plateau's physical experiments 7, 22 Plateau problem 56, 125 Poisson Simeon 7, 74 Poisson and Laplace theorem 19 Principal curvatures 20 Problem of the latest area 125 Projective space 189 Ouadruple singular point 59 Radii of curvature 65 Radiolarians 61 Rado T. 123 Reduction of the Riemannian surface 150-151 Regular minimal surface 124 Regular point 124

Relative maxima or minima 78 Representation of the minimal surface 161 Retraction 199 Riemannian surfaces 144, 146 Riemannian volume form 34 Saddle type minimal films 42 Schwarz H.A. 123 Second fundamental form 16 Segner's experiments 15 Semi-Riemannian surface 149 Shape of the drops 15 Singular chain 173 Singular cochain 192 Singular edge of the film 44 Singular points of minimal surface 55 Singular simplex 173 Skeleton figures (of the minimal surfaces) 101 Soap bubbles 9 Soap films 9 Soap films of constant positive curvature 19 Soap films of constant zero curvature 19 Stable columns of liquids 22 Stable minimal surfaces 20, 21 Stationary points 35 Steiner problem 59 Surface of constant mean curvature 16 Surface of equilibrium 16 Surface separating two media 11 Swallow tail 48, 49 Tension energy 11 Thompson D.A.W. 61 Topological genus of the surface 36 Unduloid 84 Uniqueness theorem 127 Unstable surfaces 20, 41, 42 Weierstrass representation of minimal surface 137-\38, 143 Whitney cusp 41 Whitney singularities 49 Wire contour 37 Wire models 97