The Global Theory of Minimal Surfaces in Flat Spaces: Lectures given at the 2nd Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held ... Mathematics C.I.M.E. Foundation Subseries)

. of a neighborhood of p converge on compact sets of IC to the Enneper-type minimal surface E given by g(O = ~k,

.(M) and W~ = Al+kW>.. Take R > 0 large and R' > 5R. The asymptotic geometry of E insures that the curve 1i(E n aB(R)) can be taken arbitrarily close to a (2k + I)-sheeted covering of the circle xI + x~ = 1, X3 = o. As E is unstable by Theorem 1.6, we can also assume that En B(R') is unstable. Denote by S the connected component of M~nB(R') which passes through the origin (thus S depends on A, R'). Fix A > 0 large enough so that S is extremely close to En B(R'), in particular S is unstable. As as is a nulhomologous I-cycle embedded in aw~, Theorem 1.8 insures that there exists an embedded least-area surface E C W~ with boundary aE = as (again E depends on A, R'). As E is stable but S is unstable, it follows from the maximum principle that both surfaces meet only at their boundary. As the boundary of W~ is connected and mean convex, it follows from Meeks [30] that W~ is a handle body. Therefore, E uS bounds a compact region V C W~ with piecewise smooth boundary. Note that E is no longer embedded when viewed into R 3 , because W~ immerses into ]R3. We claim that for Rand R' large enough, any component of En B(R) is embedded. This property clearly holds if for arbitrary q E En B(R), the component of EnB(5R) through q contains a graph over a disk of radius 2R: As E is stable, the length of its second fundamental form can be bounded above by 1A.~71(q) C d(q, aB(R'))-l for arbitrary q E E and some universal constant C, see Theorem 1.7. Hence, IAEI R'~5R in En B(5R). In this setting, the Uniform Graph Lemma (Lemma 4.35 below) insures that that for arbitrary q E En B(5R), the component of En B(5R) through q contains STEP

:s

:s

a graph over a disk in its tangent plane of radius r(q) = min { R'4-JR, 2R}, because the euclidean distance from q to aB(5R) is alleast 4R. If we assume from the beginning of the proof that R' > (5+8C)R, then the above minimum is not less than 2R and our claim holds. Note also that, possibly after a perturbation of R, we can suppose that E cuts aB(R) transversally, so the number of components of En oB(R) is

42

J. Perez and A. Ros

finite. Finally, the region V n B(R) is properly immersed in the ball B(R), with a finite number of boundary components. The component S n B(R) is a properly immersed closed disk. Moreover, after rescaling by the factor the boundary of this disk can be taken arbitrarily close to a (2k + 1)sheeted covering of the horizontal equator in oB(l). The remaining boundary components of VnB(R) come from portions of EnB(R) and thus all of them are embedded. By Proposition 3.23, 2k + 1 must be one. This contradiction finishes Step 3.

-k,

As a direct consequence of the Claim in Step 3 it follows that M must be an annulus and, so, the Theorem is proved. 0 Remark 3.27. The hypotheses in Theorem 3.26 can be relaxed to impose that the force of any cycle in M which is nulhomologous in W is vertical. Moreover, the surface M needs not to be embedded but only to be the (piecewise smooth mean convex) boundary of an immersed compact 3-manifold W like in Figure 3.4, see Theorem 2 in [48].

Fig. 3.4. The force along "( need not to be vertical.

Remark 3.28. The convexity of each boundary curve r i guarantees that the corresponding curve '!f1>..(ri ) remains embedded throughout the deformation. This hypothesis can be exchanged by a capillarity condition: for each i, M meets the plane containing ri with constant angle. By Lemma 3.25, this alternative hypothesis implies that '!f1>.. (ri) is homothetic to r i and hence embedded for all ).. > o. Theorem 3.26 has an interesting application to the free boundary Plateau problem, which we now describe. Suppose that r is a Jordan curve in the plane {X3 = I} and E is an immersed compact minimal surface with boundary consisting of r together a non void collection of immersed curves on a parallel plane to {X3 = I}, say II = {X3 = a}. The surface E is called a solution of

Properly embedded minimal surfaces

43

the free boundary Plateau problem with data {r, lI} if E is orthogonal to II along BE n lI. Schwarz reflection principle applies to any solution of the free boundary Plateau problem, giving rise to a minimal surface M = E U E* (the superindex * means the reflected image across lI). If we suppose additionally that r is convex, then Theorem 1 in Schoen [50] gives that E must be a graph over II (thus embedded). Meeks and White [37], also assuming the convexity of r, proved that the free boundary Plateau problem with data {r, lI} has at most two annular solutions. As the doubled surface M = E U E* of such an annular solution is a minimal annulus between two planes bounded by two convex curves, a beautiful Theorem of Schiffman [52] gives that M is foliated by convex curves in horizontal planes, thus the same holds for E. Our next statement shows that the hypothesis on the annular topology can be removed.

Corollary 3.29. Let r be a convex lordan curve in the plane {X3 = I}. Then, the free boundary Plateau problem with data {r, X3 = o} has at most two solutions, and any such solution is an embedded annulus foliated by convex curves in parallel planes. Proof. By the discussion before this Corollary, it suffices to check that any solution to the free boundary Plateau problem with data {r, lI} is an annulus. Let E be such a solution. As r is convex, E must be a graph over lI. In particular E has genus zero and, therefore, its first homology group is generated by the components of En lI. As E cuts II orthogonally, its force along any component of EnlI must be vertical. Now Theorem 3.26 joint with Remark 3.28 applies to E, concluding that it is an annulus. 0

3.3

Related Results

The above arguments can also be adapted to the case of complete embedded minimal surfaces of finite total curvature and compact boundary. The main differences reside in dealing with noncompact flat 3-manifolds instead of compact ones. This difficulty can be overcome by taking into account that such 3-manifolds consist of a compact piece (where we argue as before) together with a finite number of ends bounded by one or two representatives of annular minimal ends of finite total curvature plus a compact surface, say a portion of a ball of sufficiently large radius. The controlled asymptotic geometry of complete embedded minimal ends of finite total curvature allows to modify successfully the ideas showed in the compact case. One key difference is that we can choose between gluing planar convex disks or the exterior of these disks in the planes containing the boundary curves, to find a properly immersed flat 3-manifold W with piecewise smooth mean convex boundary. We state without proof the following result Theorem 3.30 ([48]). Let M be a properly embedded nonftat minimal surface with finite total curvature and horizontal ends. Suppose that 8M consists

J. Perez and A. Ros

44

r

of a finite number of convex Jordan curves i in parallel planes IIi, M being transversal to IIi along i , 1 :S i :S k. Let M be the piecewise smooth (immersed) surface obtained by gluing M, along each with the closure of one of the components of IIi - i , for each i = 1, ... ,k. Assume also that there exists a fiat 3-manifold W with piecewise smooth mean convex boundary M and an isometric immersion ¢ : W -+ R3 extending the immersed surface M such that ¢ embeds properly a representative of each end of W. If anyone-cycle in M which is nulhomologous in W has vertical force, then M is an annulus.

r

n,

r

As consequences of the preceeding Theorem, we point out the following statements. Corollary 3.31. There are no properly embedded minimal surfaces M C R3 with vertical forces satisfying 1. M is a global graph outside two disjoint convex disks in {X3 = O}, and 2. 8M consists of two closed convex curves in horizontal planes. Proof. Suppose M satisfies the conditions in the statement of the Corollary, with boundary components r i , r2 contained in horizontal planes IIi, II2 , respectively. The argument divides in three cases:

1: The end of M is of planar type. M is contained in the slab bounded by IIi UII2 , by the maximum principle. Define M as the piecewise smooth embedded surface obtained by gluing M along its boundary with the disk enclosed by r i in IIi and with the noncompact component of II2 - 2. Thus M bounds an embedded flat

CASE

r

3-manifold W with piecewise mean convex boundary, namely the region in the slab between M and II2 , see Figure 3.5(a)). Now Theorem 3.30 applies, hence M must be an annulus, a contradiction. CASE 2: The end is of catenoid type and the forces along r i , Tz point to the same direction (say downward pointing). As the total force along M is zero, it follows that the logarithmic growth of the end must be positive. Taking M as the union of M with the two planar disks enclosed by r i , r 2 and W as the component of R3 - M above M (see Figure 3.5(b)), we can repeat the argument before. CASE 3: The end is of catenoid type and the forces along r i , r2 point to opposite directions. We can assume that the logarithmic growth of the end is again positive, and that the height of IIi is not less than the one of II2 . From the maximum principle, M n II2 = r2. Consider M as M joint with the planar disk enclosed by r i and with the exterior of r 2 in II2 , and W as the region between M and II2 (Figure 3.5(c)), so we arrive to the same contradiction. This finishes the proof.

o

Properly embedded minimal surfaces

HI -M

Hi

/

----'-'~'

iF, \

45

'---

I (a)

-M

, (c) Fig. 3.5. The shaded zones represent the 3-manifold W enclosed by M.

The last two statements we mention as consequences of Theorem 3.30 deal with minimal surfaces without boundary, symmetric respect to a plane, say {X3 = O}. In both cases, the portion of surface in one of the halfspaces determined by {X3 = O} will satisfy the conditions in Theorem 3.30 (again exchanging the convexity of the boundary curves by the capillarity condition with angle 7r /2, see Remark 3.28). In the next Corollary, the verticality of the forces of M+ = M n {X3 > O} follows by imposing that M has genus one, thus M+ has genus zero. Corollary 3.32. There are no properly embedded genus one minimal surfaces M C R3 with horizontal ends, symmetric with respect to {X3 = O}.

In 1981, Costa [9] gave an example of a genus-one complete minimal surface with finite total curvature and three embedded ends. One year later, Hoffman and Meeks [16] proved that such surface is embedded by using that it is highly symmetric. Mathematical and computational analysis of this example allowed Hoffman and Meeks to construct, for any k ~ 1, a properly embedded minimal surface M(k) C R3 with finite total curvature, genus k and three ends [17], M(l) being the Costa surface. Moreover, they characterized M(k) by the order of its symmetry group (which is 4(k + 1)) among all surfaces with the same genus and number of ends. If one tries to extend this characterization fixing the genus but not the number of ends, then a careful analysis of the geometry of such a surface shows that we only have to discard the existence of a properly embedded minimal surface with finite total curvature, symmetric respect to {X3 = O}, such that M+ = M n {X3 > O} has genus zero and 8 M+ consists of k + 1 Jordan curves in {X3 = O}. In this setting, Theorem 3.30 gives the desired contradiction and we conclude the following Corollary 3.33. Let M C IR3 be a properly embedded minimal surface with finite total curvature and genus k > O. Then, the symmetry group of M,

46

J. Perez and A. Ros

Sym(M), has at most 4(k + 1) elements. Moreover, if ISym(M)1 = 4(k then M is, up to homothety, the surface M(k).

4

+ 1),

Limits of Minimal Surfaces

This Section is devoted to study under what conditions and in what sense we can take limits on a given sequence of minimal surfaces. This machinery is of fundamental importance in many situations as producing new examples, trapping surfaces in certain regions of space, or studying compactness questions of some moduli spaces of minimal surfaces. We develop different convergence results attending to the type of surfaces we deal with:

a) A sequence of minimal graphs, b) A sequence of minimal surfaces with local uniform bounds for the area and for the Gaussian curvature, c) A sequence of minimal surfaces with local uniform bounds for the Gaussian curvature (unbounded area), d) A sequence of minimal surfaces in an open set with local uniform bounds for the area and for the total curvature, e) A sequence of minimal surfaces in the whole ]R3 with uniformly bounded total curvature. In a recent development, Colding and Minicozzi have described the structure of limits of minimal surfaces with bounded topology and no other restriction, see [7] and references therein. 4.1

Minimal Graphs

Let J? C ]R2 be an open set and U E COO (J?). Given a multi-index a = (a, b) with a, bE NU{O}, we denote the a-th partial derivative of U by Dau = 1)~1:~¥y' where lal = a + band (x,y) E J? Thus, V'u = (D 1 u,D 2 u) is the gradient of u and 1V' 2 u1 2 = (D(1,1)u)2 + 2(D(1,2)U)2 + (D(2,2)U)2 is the squared length of its Hessian. If J?' is a relatively compact open subset of J?, we simply write J?I CC J? We endow Coo(J?) with the usual Cm-uniform topology on compacts subsets of J?, for all m ~ O. Recall that the minimal surface equation is given by (4.5)

The germ of all the results about convergence of minimal surfaces that we will see later on is the following statement for minimal graphs. Theorem 4.34. Consider a sequence {un}n C Coo (J?) of solutions of the minimal surface equation, satisfying

Properly embedded minimal surfaces

47

1. There exists p E D such that {un (p)} n is bounded. 2. {IV'unl}n is uniformly bounded on compact subsets of D. Then, there exists a subsequence {Udk C {un}n and a solution u E eOO(D) of the minimal surface equation such that {Uk h converges to u in the m _ topology, for all m.

e

Proof. The result is consequence of Corollary 16.7 in [11], reasoning as follows. Take a sub domain DI C C D such that p E D/. Hypotheses 1 and 2 together with the mean value Theorem give that {SUP!]I lunl}n is bounded. Corollary 16.7 in [11] insures that for all multi-index a, the sequence of partial derivatives {D",un}n is uniformly bounded in D/. In this situation, AscoliArzela's Theorem implies that a subsequence of {un}n converges to a function U oo E eOO(D') in em(D/), for all m. A standard diagonal process using an increasing exhaustive sequence of relatively compact domains gives a subsequence {Udk C {un}n that converges to a function U oo E eOO(D) in the em-topology in D, for all m. Clearly, U oo must also satisfy the minimal surface equation, which completes the proof. 0

As we are interested in taking limits in a sequence {Mn}n of minimal surfaces, in order to use Theorem 4.34 we need to control uniformly the relative size of the domain that expresses locally a minimal surface as a graph over the tangent plane. At this point, it is convenient to introduce some notation. Let M be a surface in IR8 with tangent plane TMp, p E M, Gauss map N : M -7 §2(1), shape operator A and Gaussian curvature K. Recall that for minimal surfaces we have IAI2 = -2K. Given p in M and r > 0, we label by D(p,r) = {p+v I v E TMp, Ivl < r} the tangent disk of radius r. W(p,r) stands for the infinite solid cylinder of radius r around the affine normal line at p, W(p, r) = {q + tN(q) I q E D(p, r), t E 1R}. Inside W (p, r) and for s

> 0, we have the compact slice

W(p,r,s) = {q+ tN(q)

I

q E D(p,r),

It I < s}.

Given an open set 0 C IR3 , we say that a minimal surface M immersed in o is properly immersed if for any relatively compact subdomain 0 1 ceO we have M n 0 1 C eM. If additionally M has no self-intersections and the topology of M is the induced by the one of 0, we will say that M is properly embedded in 0, and p.e. denote this fact simply by M CO. Lemma 4.35 (Uniform Graph Lemma). Let M be a minimal surface properly immersed in O. Suppose that IAI ~ c on M, for a given c> O. 1. For all p EM, consider R = R(p) given by

R=

min{~c' ~d(p,80)}.

(4.6)

J. Perez and A. Ros

48

Then, the component of W (p, R) n M through P is a graph over D(p, R). 2. If u E COO (D (p, R)) is the function which defines this graph, then we have the estimates lu(q)1 ~ 8clp - q12, IV'ul(q) ~ for all q E D(p,R).

8elp - ql, and IV' 2ul

~

16e,

Proof. Fix p EM. Up to a rotation, we can assume p = 0, TMp = {z = O} and N(p) = (0,0,1). As M is locally a graph, there exists a radius R > 0 with the following properties:

i) M can be expressed as the graph of a function u E COO (D(p, R)). Hence u(p) = 0 and the map 'ljJ(x,y) = (x,y,u(x,y)), (x,y) E D(p,R), is a parameterization of M with 'ljJ(p) = p. ii) The third component of the Gauss map N3 = (N, e3) = (1 + IV'uI 2)-1/2 satisfies N3 > ~ in D(p, R) (note that N 3(p) = 1). Then,

and the same is true for the derivative of N3 with respect to any unit vector. Thus IV'N31 ~ 2e. We assume that R is the maximal radius at p with the properties i), ii) above. Note that if u were defined on 8D(p, R) and N3 > ~ were true along 8D(p, R), then u could be extended to a larger radius, which contradicts the maximality of R. Hence, one of the following possibilities hold:

a) The function u extends smoothly to a larger disk and there exists q E 8D(p, R) such that N 3(q) = ~. b) There exists a sequence {qn}n C D(p, R) with d('ljJ(qn) , (0) -+ O. In case a) we get ~ = IN3(P) - N 3(q)1 ~ IV'N31(r)lp - ql ~ 2eR, where is some point in the segment [p, q]. If case b) holds, then Ip - 'ljJ(qn) I ~ length ('ljJ ([P, qn])), where [p, qn] is the segment in D(p, R) joining p and qn. We now estimate this length by

r

length ('ljJ ([P, qn])) =

10r1qnl }1 + IV'ul 2ds o < 10r1qnl 2 ds o =

21qnl < 2R,

where ds o denotes the length element in the flat disk D(p, R). So d(p, (0) ~ Ip- 'ljJ(qn) I+ d('ljJ(qn) , (0) < 2R+d('ljJ(qn), (0) -+ 2R. In summary, we obtain (4.6) which proves i). Concerning ii), firstly note that

1}1 :x1;uI21

= I(N,'ljJxx)1 = I(Nx,'ljJx)1

~ IAII'ljJxI 2~ e(1 + u;).

Therefore luxxl ~ e(1 + IV'uI 2)3/2 = eN:;3 ~ 8e. As the same holds for the other partial second derivatives, we have IV' 2ul ~ 16e in D(p, R). Using the mean value theorem we see that lux(q)1 = lux(p) - ux(q)1 ~ 8clp - ql from which one has IV'ul(q) ~ 8elp - ql. Finally, lu(q)1 = lu(p) - u(q)1 ::s; 8clp - ql2 and the proof is complete. 0

Properly embedded minimal surfaces

4.2

49

Sequences with Uniform Curvature Bounds

Bounded Area Now we formulate the notion of convergence for minimal surfaces to be studied in this Section. p.e.

p.e.

Definition 4.36. Let {Mn C O}n and M C 0 be minimal surfaces in an open set 0 C JR3. We say that {Mn}n converges to M in 0 with finite multiplicity, if M is the accumulation set of {Mn}n and for all p E M there exist 1', € > 0 such that 1. M n W (p, 1', €) can be expressed as the graph of a function u : D (p, 1') -+ R. 2. For all n large enough, MnnW(p, 1', €) consists of a finite number (independent of n) of graphs over D(p, 1') which converge to u in the em-topology, for each m ~ O.

In the situation above, we define the multiplicity of a given p E M as the number of graphs in Mn n W(p, 1', c), for n large enough. Clearly, this multiplicity remains constant on each connected component of M. Given a sequence of subsets {Fn}n in the open domain 0, its accumulation set is defined by {p E 0 I :3 Pn E Fn with Pn -+ p}. p.e.

Given a minimal surface M C 0 and a 3-ball Bee 0, we denote respectively by A(M n B) and KMnB the area and the Gaussian curvature of the portion of M inside B. Next we state our first convergence result for minimal surfaces. p.e.

Theorem 4.37. Let {Mn C O}n be a sequence of minimal surfaces. Suppose that {Mn}n has an accumulation point and that for any 3-ball Bee 0 there exist positive constants Ci = ci(B), i = 1,2, with A(Mn n B) ~ Cl and IKMnnBI ~ C2, \In E N. Then, there exists a subsequence {Mdk C {Mn}n p.e.

and a minimal surface M C 0 such that {Md k converges to M in 0 with finite multiplicity. Proof. Fix an accumulation point p of the sequence {Mn}n. Our curvature estimates assumption joint with Uniform Graph Lemma imply that there exist R = R(p) > 0 and disjoint graphs U~ C 1R3 of functions u~ defined over disks B(p,2R) n (p + (v~)J..), with Iv~1 = 1 and 1 ~ i ~ s = s(p, n), such that

= (U}, u ... u U~) n B(p, R). ii) lu~l, l'Vu~l, 1'V2u~1 are uniformly bounded in the corresponding disk of radius 2R, for all n E Nand i = 1 ... , s.

i) Mn n B(p, R)

As the area of Mn inside B(p,2R) is bounded by a constant Cl = Cl (p) > 0, we deduce that the number s of such graphs is bounded above, independently of n. Taking a suitable subsequence we can assume that s = s(p) does not depend of n and that {v~}n converges to some unit vector vi. In fact we can assume that these sequences are constant, i. e. v~ = vi, without destroying

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J. Perez and A. Ros

the derivative estimates in ii). Using Theorem 4.34, there exist subsequences {Ukh C {U~}n and minimal graphs U~ over disks or radius 2R and center P in the planes p+ (vi)J. (1 S; i S; s), such that each Uk converges to Ui . As the graphs Uk with k fixed are disjoint, maximum principle gives that each two limits graphs U i , Ui must be disjoint or coincide when restricted to B(p, 2R). If P E 0 is not an accumulation point of {Mn}, then we can choose a subsequence {Mdk and R > 0 such that Mk n B(p, R) = 0 for all k. Now take a countable dense set A = {PI,P2,"'} C O. Applying the process above around PI, we obtain a subsequence {MI,kh C {Mn}n which converges in B(Pl, R(Pl)) to a disjoint union of at most s minimal graphs with finite multiplicity. Applying again the process to {M1,k h around P2 we obtain another subsequence {M2,kh C {MI,kh which converges in B(Pl, R(PI)) U B(p2' R(P2)) to a minimal surface with finite multiplicity. Iterating the process and taking a diagonal subsequence, we obtain {Mdk C {Mn}n which converges in 0 to a minimal surface M proving the Theorem.

PC- 0

with finite multiplicity, thereby 0

Later on, we will need to identify limits of sequences of properly embedded minimal surfaces provided that the multiplicity is greater than one. p.e.

p.e.

Proposition 4.38. Let {Mn C O}n and M C 0 be minimal surfaces such that {Mn}n converges to M with finite multiplicity. If a connected component M' C M is orientable and has multiplicity m 2 2, then M' is stable. Proof. Fix a domain fl Cc M' with smooth boundary. As M' is orientable and embedded, fl has an embedded regular neighborhood fl(c) = {p+tN(p) : P E fl,ltl < c} of positive radius c with fl(c) cc 0, N being a unit normal vector field to M'. Denote by 7r : fl(c) -+ fl and d: fl(c) -+ R the orthogonal projection of fl(c) onto fl and the oriented distance to fl, respectively. From convergence of Mn to M it follows that for n large enough, 7r : Mnnfl(c) -+ fl is a m-sheeted covering map (m does not depend of n). As Mn is embedded, d must separate points at the fibers of this covering and thus Mnn fl(c) consists of m pairwise disjoint normal graphs fl1,n, ... , flm,n over fl. These sheets are naturally ordered by d and each one of them converges to fl. If we consider two consecutive sheets fll,n, fl 2,n, we can construct a narrow, half-tubular shaped, compact surface On C 0 with {JOn = {Jfll,nU{Jfl2,n, in such a way that fll,n U fl 2,n U On is a compact piecewise smooth embedded surface enclosing a 3-domain Wn Cc 0 with mean convex boundary. From Theorem 1.8, there exists a least-area surface E~ C Wn with {JE~ = {Jfl1,n' Moreover E~ is orientable. Varying n, the above procedure gives a sequence of minimal surfaces En = E~ n fl(c) properly embedded in fl(c). As {JWn collapses into fl when n -+ 00 and E~ meets (by topological reasons) the normal line TMi; for any P E fl, we deduce that the accumulation set of {En}n coincides with fl. Moreover, the stability of .En guarantees curvature estimates by Theorem 1.7. To see that {.En}n has also local area bounds, consider a 3-ball Bee J?(c) such that .En nB i= 0. Then, the area of .En nB

Properly embedded minimal surfaces

51

is not greater than the one of any piecewise smooth surface Ll c Wn with 8 Ll = 8( En n B). Note that we can construct such a surface Ll by considering suitable portions of 8B and Di,n (i = 1,2). As {Di,n}n has local area bounds because it converges to D, we conclude the desired area bounds for {En}n. Now Theorem 4.37 implies that a subsequence {Edk C {En}n converges to a minimal surface properly embedded in D(c:), with finite multiplicity. Clearly the limit surface must be D. As the area of Ek is not greater than the one of Dl ,k, we conclude that the multiplicity of E k -+ D is one. This implies easily that D is stable and concludes the proof of the Proposition. 0

Unbounded Area We will also need to construct limits of sequences of minimal surfaces under weaker conditions than in Theorem 4.37. p.e.

Theorem 4.39. Let {Mn C O}n be a sequence of minimal surfaces. Suppose that there exists a sequence Pn E Mn converging to a point p E 0 and that for any 3-ball Bee 0 there exists a positive constant c = c(B) with IKMnnBI ::; c, "In E N. Then, there exists a subsequence {Mdk C {Mn}n and a connected minimal surface M in 0 satisfying 1. M is contained in the accumulation set of {Mdk' 2. p E M and KM(p) = limk KMk (Pk). 3. M is embedded in 0 (but not necessarily properly embedded). 4. Any divergent path in M either diverges in 0 or has infinite length.

Proof. As the argument is similar to the one in Theorem 4.37, we only provide a sketch of proof. As {Pn}n accumulates at P E 0 and we have local uniform bounds for the curvature KMn around P, Uniform Graph Lemma gives R = R(p) > 0 such that the connected component of MnnB(p, 2R) passing through Pn contains a graph Un over a planar disk of center p and radius R. Moreover the functions Un which define the graphs satisfy that Iunl, IV'unl and IV' 2 uni are uniformly bounded. By Theorem 4.37 or Theorem 4.34, there exists a subsequence {Ukl h, C {Un}n converging to a minimal graph U over a disk of radius R with multiplicity one, and p E U. An analytic prolongation argument allows us to construct a subsequence {Mdk C {Mk,h, and a maximal sheet M in the accumulation set of {Mkh which extends U. By construction, the minimal surface M satisfies items 1 and 2. M must be embedded because transversal selfintersections of it would give rise to transversal selfintersections of Mn for n large, thus we have 3. Finally, take a divergent path, : [0,00[-+ M such that, does not diverge in 0 (the existence of such a curve prevents M of being proper in 0). Thus, there exists a compact set CeO and a sequence of real numbers {tili diverging to +00 such that ,(ti) E C for all i. As IKMncl is bounded, the uniform graph property implies that, up to a subsequence, there exists an interval Ii centered at ti such that I i nIHl = 0 and length(f(Ii)) > 8 for a fixed 8 > O. This gives that the length of, is infinite. 0

52

4.3

J. Perez and A. Ros

Sequences with Total Curvature Bounds

In this Section, we will exchange the local uniform curvature bounds of former results by total curvature bounds. Recall that given a minimal surface M in R 3 , its total curvature is defined by C(M) = iM /K/ dA. Limits in Open Domains We start by considering sequences of surfaces properly embedded in an open set 0 C R 3 , with local area and local total curvature bounds, see Choi and Schoen [6] and White [60]. p.e.

Theorem 4.40. Let {Mn C O}n be a sequence of minimal surfaces. Suppose that {Mn}n has an accumulation point and that for any 3-ball Bee 0 there exist positive constants Ci = ci(B) > 0, i = 1,2, with A(Mn n B) :::; Cl and C(Mn n B) :::; C2, 'tin E N. Then, there exists a subsequence {Mdk C {Mn}n,

a discrete set X C 0 and a minimal surface M converges to M in 0 - X with finite multiplicity. Moreover, given P E X and R > 0 it holds lim sup C(Mk

n B(p, R)) 2

p.e.

C

0 such that {Mdk

47L

(4.7)

k

In what follows, we will call X the singular set of the sequence {Mdk. Proof. Define X = {p EO / {/KMnnB(p,r)/}n is unbounded, 'tIr > O}. Fix a point P E X and take a radius l' > 0 with B(p,r) CC O. Let Pn be a maximum of the function /KMn (- )/d( -, 8B(p, 1'))2 in the closure of Mn n B(p, 1') (observe that this function is invariant under rescaling). Define the sequences An = V/KMn(Pn)/ and rn = d(Pn,/)B(p, 1')). As /K MnnB (p,r/2)/ is unbounded, after passing to a subsequence we find points qn E Mn n B(p,r/2) with /KMn(qn)/ -+ +00. Note that A;r; 2 /K Mn (qn)/d(qn,8B(p,r))2 2 /KMn(qn)/r:, thus {Anrn}n also diverges to 00. Translate Pn to the origin and homothetically expand Mn n B(Pn, rn) by the factor An, so we obtain new minimal surfaces Mn PC- B(O, Anrn) passing through the origin (see Figure 4.1), whose curvatures satisfy /KMn (0)/ = 1 for all n. Given R

> 0 and q E Mn n B(O, R),

/KMJ(f)/(Anrn - R)2:::; /K MJ(f)/d(q,8B(0, Anrn))2 = /KMn (q)/d(q, 8B (Pn , rn))2 :::; /KMn (q)/d(q, 8B(p, 1'))2 :::; A~r~, where q E Mn is the point which corresponds to q E M through the rescaling. Thus we get that {/K Mn /}n is uniformly bounded on compact subsets of R3. Note also that the invariance of the total curvature under rescaling shows that C(Mn) is bounded above by a constant that only depends on r. Therefore there exists a subsequence {Mdk C {Mn}n and a complete nonflat minimal surface M C R3 such that Mk converges (in the sense of Theorem 4.39) to

Properly embedded minimal surfaces

Fig. 4.1. Rescaling Mn

n B(Pn, Tn)

53

by factor An

M. Moreover, it is clear that M has finite total curvature. As M is nonflat, it must have total curvature at least 47r, by (1.3). Hence, coming back to the original scale, we deduce that lim sup C(Mk n B(p, r)) ~ 47r. This property, joint with the uniform control of the total curvature, imply that X is discrete. By definition of X, {Kn}n is uniformly bounded on compact subsets of o-x. As we have local area bounds, Theorem 4.37 insures that a subsequence of {Mn}n converges to a minimal surface M PC- 0 - X with finite multiplicity. It only remains to prove that M can be extended to a properly embedded minimal surface in O. This fact will we a consequence of Lemma 4.42 below.

o Remark 4.41. In the above proof we saw how to produce, around a singular point p E X and after rescaling, a nonflat minimal surface M with finite total curvature. As the extended Gauss of such a surface must be onto, we deduce that the Gauss map of the surfaces Mk n B(p, R) cannot be contained in an open hemisphere.

Lemma 4.42. Let Me B(O, 1)-{O} be a properly embedded minimal surface with compact boundary contained in {Ipl = 1}. If M has finite total curvature, then M extends through the origin giving rise to a properly embedded minimal surface in B(O, 1). Proof. We first show that M is conformally a compact Riemann surface with boundary minus a finite number of points. Let f : ]R3 - {O} -+ ]R3 - {O} be the inversion given by f(p) = ~, P E ]R3 - {O}. As f is a conformal

diffeomorphism, we have that M = f(M) is a properly embedded (nonminimal) surface in the exterior of the unit 3-ball, with boundary contained in {Ipl = I}. In particular, M is complete. The relationship between the induced metric ds 2 by the inner product in ]R3 and the pullback metric ([82 = 1*(,) is

54

cJ:S2 =

J. Perez and A. Ros

Ipl- 4 ds 2 , and the respective curvature elements are related by

N being the Gauss map of M. In particular,

::; - r _

i{K
K dA ::; -

r K dA = C(M) < +00.

iM

Hence Huber's Theorem [19,59] implies that M has the conformal structure of a compact Riemann surface with boundary with a finite number of points removed, Pi, ... ,Pk' Thus for r > 0 small enough, M(r) = MnB(O, r) is a union of properly embedded minimal surfaces D*(Pl),"" D*(pk) C B(O, r) - {O} which are conform ally equivalent to punctured disks and whose boundaries are contained in {ipi = r}. As the coordinate functions on these surfaces are harmonic and bounded by r, they can be extended through the punctures Pi so they produce minimal (possibly branched) disks D(pd, ... , D(Pk). As a minimal surface must have selfintersections in any neighborhood of a branch point, it follows that none of the D (Pi) is branched. Finally, k must be one because otherwise we would have two minimal disks touching only at an interior point, in contradiction with the maximum principle. Now the Lemma is proved. 0

Limits in R3 In the second part of this Section, we deal with sequences of surfaces properly embedded in the whole space. In this setting, to take limits we only require total curvature bounds. Theorem 4.43 ([47]). Let {Mn PC'R3}n be a sequence of minimal surfaces with fixed finite total curvature C(Mn) = c for all n. Then, there exists a subsequence {Mkh C {Mn}n such that one of the following possibilities hold: 1. {Mdk has no accumulation points. 2. There exists a finite set X C R3 such that {Mk} k converges in R3 - X to a finite union of parallel planes, with finite multiplicity. Moreover, equation (4.7) holds at any point of the singular set X.

3. There exists a minimal surface M PC' R3 such that C(M) ::; c and {Mdk converges to M in R3 with multiplicity one. Proof. Assume that {Mn}n has an accumulation point. As the total curvature of the surfaces Mn is fixed, formula (1.3) implies that the number of ends of any Mn is bounded above by a fixed integer r 2:: 2. Thus, Proposition 1.5

Properly embedded minimal surfaces

55

insures uniform local area bounds for the sequence {Mn}n. Under these conditions, Theorem 4.40 says that there exist a subsequence {Mkh c {Mn}n and a minimal surface M PC-]R3 such that {Mdk converges to M in]R3 minus a discrete set X with finite multiplicity. As C(Mk ) = c for all k, it follows that C(M) is finite and not bigger than c. Moreover, the inequality (4.7) for arbitrary p E X together with the hypothesis C(Mk ) = c insure that X is finite. Note that as M is properly embedded in ]R3, it must be connected or a union of parallel planes by the strong halfspace Theorem in [18]. In this last case, the inequality A(Mk n B(p, R)) ::; r-rr R2 of Proposition 1.5 implies that M is a union of at most, r parallel planes. It only remains to prove that if M is nonftat (hence connected), the multiplicity of the limit {Mdk -+ M is one and the singular set X is empty. Reasoning by contradiction, suppose that {Mdk -+ M has multiplicity greater than one. Firstly note that as M is properly embedded in ]R3, it must be orient able and the same holds for M - X. As the multiplicity of the limit {Mk} k -+ M - X is at least 2, Proposition 4.38 implies that M - X is stable. But M extends smoothly through each p EX, and a standard cutoff functions argument insure that the property of its Jacobi operator being positive semidefinite in a punctured surface extends to the whole surface. Thus M is a stable minimal surface properly embedded in ]R3. Then Theorem 1.6 implies that M is a plane. This contradiction shows that the multiplicity of the limit must be one. Finally, take a singular point p EX. Choose r, c > 0 small such that X n W(p,r,c) = {p} and M n W(p,r,c) is a graph over the tangent disk D(p, r), say of a function u. As Mk is proper, MknW(p, r, c) must be compact for all k. Moreover, the convergence with multiplicity one of {Mdk to M in ]R3 - X insures that for k large enough, Mk n 8W (p, r, c) is the graph of a function Vk : 8D(p, r) -+ ]R with Vk -+ u in cm (8D(p, r)) for all m ~ O. As Mk n W(p, r, c) is compact, Proposition 1.1 insures that this surface is indeed a graph over D(p, R). In particular, the Gauss map of Mk n W(p, r, c) is contained in an open hemisphere, which contradicts that p is a singular point, see Remark 4.41. This finishes the proof of the Theorem. 0

5

Compactness of the Moduli Space of Minimal Surfaces

Given integers g ~ 0 and r ~ 1, we will denote by M(g, r) the space of properly embedded minimal surfaces in ]R3 with genus g and r horizontal ends. As we saw in Subsection 1.2, M(g, 1) is empty when g ~ 1, while M(O, 1) is just the space of horizontal planes. Theorem 1.3 and Corollary 2.13 say that M(0,2) consists only of Catenoids, and that M(g, 2), M(O, r) are empty for g ~ 1, r ~ 3.

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Choi and Schoen [6] have proved that the space of embedded compact minimal surfaces of fixed genus in the standard unit 3-sphere §3(1) is compact, in the sense that given any sequence in this space we can find a subsequence which converges smoothly to a minimal surface in S3 (1) with the same topology. In our setting, a natural question is to decide whether the moduli space M(g,r) is compact, that is, if any sequence {Mn}n C M(g,r) has a subsequence which converges (up to homotheties) to a minimal surface M E M(g,r) with multiplicity one (in what follows, homothety stands for a homothety or a translation in IR 3). The spaces M(O, 1) and M(O, 2) are compact but M(l, 3) is known to be noncompact. In fact, M(l, 3) is the only nontrivial nonvoid moduli space which has been completely described: Costa proved in [10] that M(1,3)/{homotheties} = R In this section we will see that for some prescribed topologies, the moduli space M(g, r) is compact (this holds, for instance, when g = 1 and r = 5). The results below were obtained by Ros in [47]. A central open problem in our setting is a conjecture by Hoffman and Meeks [17], which asserts that for each genus g ~ 1, there exists an integer r(g) such that M(g,r) is empty for r > r(g) (more precisely, they conjecture that r(g) = g + 2). The compactness result above may be viewed as a first step in the proof of the Hoffman and Meeks problem: what we expect is that these compact moduli spaces are in fact empty. 5.1

Weak Compactness

In this Subsection we study convergence of sequences of surfaces in a fixed space M (g, r). Roughly speaking, we show that any sequence {Mn} n C M (g, r) must have a partial that converges in an appropriate sense to a finite collection of surfaces Mi,oo E M(gi,ri), 1 ~ i ~ k, with gi ~ g and ri ~ r. As we cannot insure the convergence with multiplicity one to a single surface in the original space M (g, r), we will use the expression weak compactness to refer to this property. Recall that for M E M (g, r), its total curvature depends only on g and r, see equation (1.3). ~ 0, r ~ 2. Given a sequence of surfaces {Mn}n C M(g,r), there exist a subsequence, denoted again by {Mn}n, an integer k > 0, a collection of nonfiat minimal surfaces Mi,oo E M(gi, ri) with gi ~ g, ri ~ r for i = 1, ... , k, and k sequences of homotheties {hi,n}n satisfying

Theorem 5.44. Fix integers g

1. C(Mi,oo) + ... + C(Mk,oo) = C(Mn ), 2. {hi,n(Mn)}n converges to Mi,oo in IR3 with multiplicity one, 1 ~ i ~ k, 3. For any R, n large, there exist k disjoint balls B1,n," . ,Bk,n C ]R8 with hi,n(Bi,n) = B(O, R) such that Mn decomposes as

Mn = M1,n U ... U Mk,n U fh,n U ... U flr,n,

Properly embedded minimal surfaces

57

where Mi,n = MnnBi,n (hence hi,n(Mi,n) can be taken arbitrarily close to Mi,oo n B(O, R) for n large enough) and [lj,n is a graph over the exterior of some convex disks in {X3 = O}, containing exactly one end of Mn.

In this setting, we will call Ml,n, ... ,Mk,n the bounded domains, [ll,n, ... , [lr,n the unbounded domains of the surface Mn (see Figure 5.1) and Ml,oo, ... , Mk,oo the weak limit of the subsequence {Mn}n. Proof. Equation (1.3) gives that the

_ (:-~~~.:'-- -fl2.n Ie,-_

Fig. 5.1. A surface with five ends decomposed in bounded and unbounded domains.

total curvature of all the Mn is fixed, say C(Mn) = c. Given n E N, consider balls B C R3 with C(Mn n B) = 27r. As r 2: 2, we have that Mn is nonflat and so, c 2: 47r. In particular, the above family of balls is nonvoid. Clearly if the center of a ball in the family goes to infinity, then its radius must also diverge to infinity. Thus we can find a ball B~ n in this family with minimum radius. Let hl,n be the homothety that transforms BLn into B(O, 1). All the rescaled surfaces {hl,n(Mn)}n have total curvature c. By Theorem 4.43, there exists a subsequence, again denoted by {Mn}n, such that one of the following possibilities hold: a) there exists a finite set Xl C R3 such that {hl,n(Mn)}n converges in R3 - Xl to a finite union of parallel planes with finite multiplicity, or b) there exists a minimal surface Ml,oo PC- R3 such that C(Ml,oo) {hl,n(Mn)}n converges to Ml,oo in R3 with multiplicity one.

:S c and

Let us see that case a) is impossible. Reasoning by contradiction, take a point p in the singular set Xl. Equation (4.7) at p implies that we can find a ball

B(p, R) of arbitrarily small radius such that C(hl,n(Mn) n B(p, R)) 2: 37r. As hl,n(BLn) = B(O, 1), the existence of B(p, R) contradicts the minimality of the radius of BLn. As consequence, only case b) can hold. In particular, C(Ml,oo n B(O, 1)) must be 27r and so, M1,oo is nonflat. Denote by gl,rl

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J. Perez and A. Ros

the genus and number of ends of M 1 ,oo, respectively. From the convergence {h 1,n(Mn n n -+ M 1,oo it follows that gl :::; g. Finally, consider a large positive number P1 such that IC(M1 ,oo) - C(h 1,n(Mn) n B(O,pt))1 < e, with e > small. Note that P1 can be chosen so that h 1,n(Mn ) n 8B(O, P1) consists of r1 Jordan curves projecting bijectively onto convex curves in the limit tangent plane to M 1 ,oo (which at this moment need not to be horizontal, although this will certainly be the case). Denote by B 1,n = hl,;(B(O,P1))' This finishes the first step in our construction of the weak limit. Assuming C(M1 ,oo) < c, we will construct the second partial limit of our sequence. As both C(M1 ,oo) and c are integer multiples of 471", the family of balls B C R3 such that C([Mn - B 1,n] n B) = 271" is nonvoid. As before, we can choose a ball B~,n in this family with minimum radius. Clearly the radius of B~,n cannot be smaller than the radius of BLn" We label as h 2 ,n the homothety such that h2,n(B~,n) = B(O, 1). Thus, the radius of h 2,n(BLn) is at most one, and the one of h 2,n(B 1,n) is bounded above by Pl. We claim that if {h 2 ,n(B1,nnn has a limit, then it must be necessarily a single point: by contradiction, assume that {h 2 ,n(B1,n)}n converges to a ball of positive radius. Then, the surfaces h 1,n(Mn ) and h 2 ,n(Mn ) differ in a homothety whose center and ratio are controlled independently of n. As {h 1,n(Mn n n converges in R3 to M 1,oo with multiplicity one, we deduce that C(h 2,n(Mn ) n [B(O, 1) - h 2 ,n(B1,n)]) can be made arbitrarily small, which contradicts our choice of B~,n' Hence our claim holds. Using again Theorem 4.43 and after passing to a subsequence, we have two possibilities: either {h 2 ,n(Mn n n converges in R3 minus a finite subset X 2 to a finite union of parallel planes with finite multiplicity, or {h 2,n(Mn n n converges in R3 to a properly embedded minimal surface M 2 ,oo with multiplicity one, C(M2 ,oo) being less that or equal to c. Our next goal is showing that the singular set X 2 must be empty and only the second possibility can occur. On the contrary, if X 2 :j:. 0 then the minimizing property of B~,n implies that X 2 = {p} = limn h 2,n(B1,n). Moreover, p E B(O,I) (otherwise for n large we would have h 2 ,n(B 1,n) n B(O, 1) = 0 thus {h 2 ,n(Mn ) n B(O, Inn would converge in B(O, 1) to a finite union of parallel disks with finite multiplicity, which contradicts that C(h 2 ,n(Mn ) n B(O, 1)) = 271" for all n). Fix e > small. Taking n large enough, we can suppose h 2 ,n(B1 ,n) CC B(p,e). Then

°

°

271" = C ([h 2 ,n(Mn ) n B(O, 1)]- h 2 ,n(B 1,n)) =

C ([h 2 ,n(Mn ) n B(O, 1)]- B(p, e))

+ C (h 2 ,n(Mn -

B 1,n) n B(p, e)). (5.8)

As {h 2 ,n(Mn n n converges in R3 - {p} to a finite union of parallel planes with finite multiplicity, the first summand in (5.8) goes to zero as n -+ 00. Take a component S of h 2 ,n(Mn - B 1,n) n B(p, e). The components of the boundary of S are divided in two kinds: the ones which lie on 8B(p,e), where the Gauss map of S converges to a constant value of the sphere §2(1) because outside p the limit of {h 2 ,n(Mn)}n is fiat, and those lying on 8h 2 ,n(Bl,n), where the

Properly embedded minimal surfaces

59

Gauss map of S is again almost constant because of the existence of the limit surface MI,oo. As the Gauss map of a nonflat minimal surface is an open map, all these constants in §2(1) must be the same (otherwise the Gauss map image of S would cover almost all §2(1), hence C(S) would be close to a positive multiple of 47[, a contradiction with the definition of B~,n). Equivalently, the Gauss map image of S is contained in a small neighborhood of some vector a E §2(1). Moreover, ±a must coincide with the limit normal vectors at the ends of MI,oo and with the normal vectors to the flat limit of {h 2,n(Mn )}n outside p. Clearly, C(S) can be taken arbitrarily small by choosing n large enough. Note also that the number of such components S is bounded above independently of n, because the number of boundary components of h 2,n(Mn -BI,n)nB(p, c) is controlled by the (finite) number of planes in the flat limit of {h 2,n(Mn )}n and the number TI of ends of MI,oo. As consequence, the second summand in (5.8) will be also arbitrarily small for n large, which is the desired contradiction. So we have proved that X 2 = 0 and, therefore, {h 2,n(Mn )}n converges in ]R3 to a properly embedded minimal surface M 2 ,oo with multiplicity one. As before, C(M2,oo n B(O, 1)) = 27[ hence M 2,oo is nonflat. Calling g2, T2 to its genus and number of ends, it follows that g2 ::; g. Take P2 > large such that IC(M2 ,oo) - C(h2,n(Mn ) n B(O,P2))1 < c and h 2,n(Mn ) n aB(O,P2) consists of T2 Jordan curves projecting bijectively onto convex curves in the limit tangent plane to M 2,oo. Finally, denote by B 2,n = h2,;(B(O,P2)). Recall that the radius of h 2,n(B I ,n) is at most Pl. This inequality together with the fact that X 2 = 0, imply that the sequence of balls {h 2 ,n(B I ,n)}n must diverge in ]R3. In particular, we can assume that h 2,n(B I ,n) n B(O, P2) = 0 and so, BI,n, B 2,n are disjoint. Now our second step is finished. Clearly, C(MI,oo) + C(M2 ,oo) ::; c. If the equality does not hold, then we repeat the arguments above and in a finite number of steps, say k, we reach the equality. To finish the proof, we must prove that each limit surface Mi,oo has Ti ::; T horizontal ends and Mn decomposes as in item 3 of the statement. With this aim, let Dn be the closure of a component of Mn - (BI,nU . . .UBk,n). Dn is a properly embedded minimal surface in ]R3 - (BI,n U ... U Bk,n) whose boundary consists of a finite number of Jordan curves in the boundaries of some of the balls Bi,n. As the homothetical expansion of Mn n Bi,n by hi,n is arbitrarily close to the intersection of a big ball with the properly embedded nonflat minimal surface of finite total curvature Mi,oo, it follows that each component rn of aDn, say in aBi,n, is close to a round circle for n large and the Gauss map of Mn along rn is uniformly close to a constant value, namely the limit normal vector of the corresponding end of Mi,oo. In particular, the Gauss map of Dn applies aDn into curves contained in small neighborhoods of some constants values of §2(1). As such Gauss map is an open map and the total curvature of Dn is small, it follows that all these constants are the same. In other words, the Gauss map image of Dn is contained in a small neighborhood of a vector a E §2(1). Along each component rn c aBi,n of aDn, glue Dn smoothly with a compact surface Di,n C Bi,n, Di,n being a graph over the orthogonal plane (a)-L to a (we can take such Di,n close to a planar disk

°

J. Perez and A. Ros

60

parallel to (a)l.). After these gluing processes, we obtain a properly embedded (nonminimal) surface n~ without boundary, whose Gauss map image is contained in a small neighborhood of a. The projection of n~ over (a)l. is a proper local diffeomorphism, thus a covering map and then necessarily a global diffeomorphism. In particular, nn is graph over a noncompact region in (a)l. bounded by a finite number of disjoint convex curves. This implies that nn contains exactly one end of Mn, and that a must be the value of the Gauss map of Mn at this end, a = ±(O, 0,1), so item 3 of the statement is true. Note also that all the limit surfaces Mi,oo corresponding to the balls Bi,n joined to a given unbounded domain nn along components of ann must have horizontal limit tangent plane. As all the limit surfaces Mi,oo will appear when considering all the unbounded domains, we conclude that Mi,oo has horizontal ends for all i. To finish the proof, we check that the number of ends r i of Mi,oo is less than or equal to r: note that the boundary components of Mn n Bi,n correspond bijectively with the ends of Mi,oo. Moreover, MnnBi,n is joined along each one of these components to a certain unbounded domain, which contains exactly one end of Mn. So we have an injective map from the ends of Mi,oo into the ends of Mn. Then ri ~ r and we have proved the Theorem. 0 Remark 5.45. Item 1 in Theorem 5.44 joint with equation (1.3) lead us to k

k

i=l

i=l

L gi + L ri 5.2

k = g + r - 1.

(5.9)

Strong Compactness

Next we prove the compactness results stated at the beginning of this Section. Firstly we study more carefully the unbounded domains appearing in Theorem 5.44. Following the above notation, take a sequence {Mn}n C M(g,r) weakly convergent to M1,oo, ... , Mk,oo. Decompose each Mn with n large in bounded and unbounded domains Mn = M1,n U ... U Mk,n U n1,n U ... U nr,n. Both the set P = {n1,n, ... , nr,n} of unbounded domains and the set of boundary components of bounded domains are naturally ordered by their heights with respect to the vertical direction. Fix n E P (note that an ::j:. 0). A component r of an is said to be a top boundary component if r is the top boundary component of the bounded domain Mi,n which contains r. Bottom boundary components of n are defined similarly. Clearly, the top (resp. bottom) unbounded domain only contains top (resp. bottom) boundary components. We decompose P as the disjoint union of the following three sets:

A = {the top and bottom unbounded domains}, B = {n E P - A / n has only top or bottom boundary components}, C = P - (AUB).

Properly embedded minimal surfaces

61

Proposition 5.46. If [l E B, then [l has at least three boundary components. Proof. Fix [l E B. Firstly suppose that a[l is connected. As [l E B, a[l must be the top or boundary component of a bounded domain Mi,n C Bi,n joined with [l along its boundary. Hence we can find an open disk D C Bi,n with aD = a[l and Mi,n n D = 0. Note that [l U D is a properly embedded topological plane, so it separates R3 in two components. As Mn is disjoint with D, it follows that Mn is contained in the closure of one the of the components of R3 - ([l U D), thus [l is the top or the bottom unbounded domain, a contradiction. So, [l must have least two boundary components. Assume now that [l has exactly two boundary components r 1 , r 2 . For i = 1,2, [l is joined along ri to a bounded domain Mi,n, ri being a top or bottom boundary component of Mi,n. As n ---+ 00, a suitable rescaling of Mi,n converges by Theorem 5.44 to the intersection with a big ball of one of the surfaces Mi,oo in the weak limit. As the top and bottom ends of each Mi,oo are of Catenoid type, we can suppose by taking n large enough that near each ri , [l looks like a neighborhood of a horizontal section in a vertical halfcatenoid. Cutting transversally [l with suitable horizontal planes, we obtain a proper subdomain [ll C [l whose boundary consists of two convex Jordan curves r{ ,r~ in horizontal planes. Moreover [ll is close along r: to the intersection of a vertical halfcatenoid with a horizontal slab, i = 1,2. If we prove that the force of [ll along each is vertical we will contradict Corollary 3.31, thereby finishing the proof of the Proposition. To prove that the forces along r{, r~ are vertical, it suffices to check that each one of these curves disconnects Mn (in such case each would be homologous to a sum of curves around the ends of M n , whose forces are vertical). Let D i be the planar convex open disk bounded by Thus M n n D i = 0 for n large, and [ll U Dl U D2 is again a properly embedded topological plane hence it separates R3 in two components N 1 , N 2 . As [l is neither the top or the bottom unbounded domain of M n , we deduce that Mn meets Nl and N 2 . If both r{, r~ bounded top ends of M1,oo, M 2 ,oo (i.e. if the corresponding curves r1,n were top boundary components of M1,oo and M 2 ,oo) , then Mn - [ll would be below [ll U Dl U D2 in a neighborhood of r{ U r~, hence Mn - [ll would be entirely contained in the component N j below [l/UDl UD 2 , which is impossible. Similarly, both r;, r~ cannot bound bottom ends of M1,oo, M 2 ,oo, thus one bounds a top and the other bounds a bottom end. This implies that one component of a[ll, say r;, is the topological boundary of Mn n Nl and r~ is the one of Mn n N 2 . In particular, each one of these curves disconnects M n , which finishes the proof. 0

r:

r:

r:.

Let #(P) and #(Pa) be respectively the number of elements in P and in P a = {boundary components of elements in Pl.

We use similar notations with A, B, C instead of P. Clearly #(P) = r, #(Pa) = Eiri, #(A) = 2 and #(Aa)::::: 2. As A,B,C form a partition

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J. Perez and A. Ros

of P, we get

r = 2 + #(B)

+ #(C).

(5.10)

On the other hand, each n E C has at least a boundary component which is neither a top nor a bottom boundary component. As each one of the k surfaces Mi,oo in the weak limit has ri - 2 middle ends, it holds #(C) ~ L~=l ri - 2k, and thus, k

r ~ 2 + #(B)

+ L ri -

2k.

(5.11)

i=l

Proposition 5.46 insures that 3#(B) ~ #(Ba). Note that the total number of top and bottom ends in all surfaces Mi,oo is 2k, and at least two of them correspond to the elements of A (because the top and bottom unbounded domains only contain top and bottom boundary components, at least one each). Thus #(Ba) ~ 2k - 2 and

#(B)

~

2k; 2.

To finish these general counting arguments, from #(Pa) = #(Aa) #(Ca) and 3#(B) ~ #(Ba) one has

(5.12)

+ #(Ba) +

k

L ri ~ 2 + 3#(B) + #(Ca).

(5.13)

i=l

Next result, which deals with the case where the weak limit has the simplest topology, is related with the Hoffman-Meeks conjecture. Theorem 5.47. Fix integers g ~ 0, r ~ 2. If a sequence {Mn}n C M(g, r) is weakly convergent to M1,oo,"" Mk,oo and each Mi,oo is a Catenoid, then r ~ 2g + 2.

Proof. We continue with the same notation as before. As each Mi,oo is a Catenoid we have gi = 0, ri = 2 (1 ~ i ~ k) and all the boundary components of any unbounded domain are top or bottom boundary components, which implies C = 0. Thus equations (5.10) and (5.13) transform respectively into r = 2 + #(B) and 2k ~ 2 + 3#(B). These two relations give 2k ~ 3r - 4. On the other hand, (5.9) gives k = 9 + r - 1, from where r ~ 2g + 2 follows 0 directly.

If in the statement of Theorem 5.44 we have k = 1, then {Mn}n converges, up to homotheties, to a properly embedded minimal surface Moo with multiplicity one, without loss of total curvature or topology. The following result gives a condition to insure such kind of strong convergence. Theorem 5.48. Fix integers 9 ::::: 0, r ::::: 2. If a sequence {Mn}n C M(g, r) is weakly convergent to M1,oo, ... , Mk,oo and M1,oo has genus g, then k = 1.

Properly embedded minimal surfaces

63

Proof. As gl + .. .+gk :::; g and gl = g, for i ~ 2 we get gi = 0 hence ri = 2 by Corollary 2.13. To compute r1 we use (5.9), which now gives r = rl + k - 1. Hence (5.11) writes as r1 + k - 1 :::; 2 + #(B) + 2::7=1 ri - 2k = #(B) + rl, i.e. #(B) ~ k - 1. This inequality joint with (5.12) force k to be one. 0

If any sequence in M(g, r) has a subsequence converging to a weak limit with k = 1, we will say that M(g, r) is compact (up to homotheties). Finally we prove two results about this property.

Corollary 5.49. For any r

~

5, the space M(l, r) is compact (up to homo-

theties). Proof. Fix r ~ 5 and take a sequence {Mn} n C M (g, r) weakly convergent to M1,oo, ... , Mk,oo. The Corollary will be proved if we check that k = 1. On the contrary, if k ~ 2 then Theorem 5.48 insures that all the Mi,oo have genus zero. Using Theorem 5.47, we must have r :::; 4, a contradiction. 0

As a generalization of Corollary 5.49, we have Corollary 5.50. Fix integers g ~ 1, r > 2g + 2. If M(g',r') = 0 for each pair (g', r') with 0 :::; g' < g and r' > 2g' + 2, then M(g, r) is compact (up to homotheties) . Proof. Let {Mn}n C M(g,r) be a sequence of minimal surfaces which converges weakly to M1,oo, ... , Mk,oo. Reasoning by contradiction, assume k ~ 2. By Theorem 5.48, the genus gi of each Mi,oo must be strictly less than g, hence our hypotheses imply ri :::; 2gi + 2, ri being the number of ends of Mi,oo. Plugging this inequality and r > 2g + 2 into (5.9) we get 3(2:: i ri - r) < 4(k - 1), which joint with (5.11) gives 2k - 2 < 3#(B), in contradiction with (5.12). 0

References 1. E. Calabi, Quelque applications de l'Analyse complex aux surfaces d'Aire minimal, Topics in Complex Manifolds, Les Presses de l'Universite de Montreal (1968). 2. M. P. do Carmo & C. K. Peng, Stable complete minimal surfaces in ]R3 are planes, Bull. Amer. Math. Soc. 1 (1979) 903-906. 3. C.C. Chen & F. Gackstatter, Elliptische und hyperelliptische funktionen und vollstandige minimalfiachen vom Enneperschen type, Math. Ann. 259(3) (1982) 359-369. 4. J. Choe & M. Soret, Nonexistence of certain complete minimal surfaces with planar ends, preprint. 5. H. Choi, W. H. Meeks III & B. White, A rigidity theorem for properly embedded minimal surfaces in ]R3, J. Differ. Geom. 32 (1990) 65-76. 6. H. Choi & R. Schoen, The space of minimal embeddings of a surface into a three dimensional manifold of positive Ricci curvature, Invent. Math. 81 (1985) 387-394.

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J. Perez and A. Ros 7. T. H. Colding & W. P. Minicozzi II, Minimal surfaces, Courant Institute of Mathematical Sciences Lecture Notes 4 (1999). 8. P. Collin, , CIME proceedings, Springer-Verlag (2000). 9. C. Costa, Example of a complete minimal immersion in R3 of genus one and three embedded ends, Bul!. Soc. Bras. Mat. 15 (1984) 47-54. 10. C. Costa, Uniqueness of minimal surfaces embedded in R3 with total curvature 127r, J. Differ. Geom. 30(3) (1989) 597-618. 11. D. Gilbarg & N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edit. Springer-Verlag (1983). 12. Goursat, Sur un mode de transformation des surfaces minima, Acta Math. 11 (1887-8) 135-86. 13. D. Fischer-Colbrie & R. Schoen, The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature, Comm. Pure App!. Math. 33 (1980) 199-211. 14. D. Hoffman & H. Karcher, Complete embedded minimal surfaces of finite total curvature, in 'Geometry V', Encyclopaedia of Math. Sci. 90 (R. Osserman, ed.), Springer-Verlag (1997) 5-93. 15. D. Hoffman, H. Karcher & F. Wei, The genus one helicoid and the minimal surfaces that led to its discovery, Global Analysis and Modern Mathematics, Karen Uhlenbeck, editor, Publish or Perish Press (1993) 119-170. 16. D. Hoffman & W. H. Meeks III, A complete minimal surface in R3 with genus one and three ends, J. Diff. Geom. 21 (1985) 109-127. 17. D. Hoffman & W. H. Meeks III, Embedded minimal surfaces of finite topology, Ann. Math. 131 (1990) 1-34. 18. D. Hoffman & W. H. Meeks III, The strong halfspace theorem for minimal surfaces, Invent. Math. 101 (1990) 373-377. 19. A. Huber, On subharmonic functions and Differential Geometry in the large, Comm. Math. Helv. 32 (1957) 13-72. 20. L. Jorge & W. H. Meeks III, The topology of complete minimal surfaces of finite total Gaussian curvature, Topology 2 (1983) 203-221. 21. N. Kapouleas, Complete constant mean curvature surfaces in Euclidean three-space, Ann. Math. 131 (1990) 239-330. 22. N. Kapouleas, Complete embedded minimal surfaces of finite total curvature, J. Diff. Geom. 47(1) (1997) 95-169. 23. N. Korevaar, R. Kusner & B. Solomon, The structure of complete embedded surfaces with constant mean curvature, J. Diff. Geom. 30 (1989) 465-503. 24. R. Langevin & H. Rosenberg, A maximum principle at infinity for minimal surfaces and applications, Duke Math. J. 57(3) (1988) 819-828. 25. H. B. Lawson, Jr. Lectures on minimal submanifolds, Publish or Perish Press, Berkeley (1971). 26. F. J. Lopez, New complete genus zero minimal surfaces with embedded parallel ends, Proc. A. M. S. 112 (1991) 539-544. 27. F. J. Lopez & F. Martin, Complete minimal surfaces in R 3 , Publicacions Matematiques 43 (1999) 341-449. 28. F. J. Lopez & A. Ros, On embedded complete minimal surfaces of genus zero, J. Diff. Geom. 33 (1991) 293-300.

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29. R. Mazzeo & D. Pollack, Gluing and moduli for noncompact geometric problems, Geometric Theory of Singular Phenomena in PDE, Symposia Mathematica vol. XXXVIII Cambridge Univ. Press, July (1998). 30. W. H. Meeks III, Lectures on Plateau's problem, Instituto de Matematica Pura e Aplicada (IMPA), Rio de Janeiro, Brazil (1978). 31. W. H. Meeks III, The theory of triply periodic minimal surfaces, Indiana Univ. Math. J. 39 (1990) 877-936. 32. W. H. Meeks III, The geometry, topology and existence of periodic minimal surfaces, Proc. of Symposia in Pure Mathematics 54(2) AMS (1993) 333-374. 33. W. H. Meeks III, J. Perez & A. Ros, Uniqueness of the Riemann minimal examples, Invent. Math. 131 (1998) 107-132. 34. W. H. Meeks III & H. Rosenberg, The global theory of doubly periodic minimal surfaces, Invent. Math. 97 (1989) 351-379. 35. W. H. Meeks III & H. Rosenberg, The maximum principle at infinity for minimal surfaces in fiat three manifolds, Comm. Math. Helv. 65(2) (1990) 255-270. 36. W. H. Meeks III & H. Rosenberg, The geometry of periodic minimal surfaces, Comm. Math. Helv. 68 (1993) 538-578. 37. W. H. Meeks III & B. White, Minimal surfaces bounded by convex curves in parallel planes, Comm. Math. Helv. 66(2) (1991) 263-278. 38. W. H. Meeks III & S. T. Yau, The existence of embedded minimal surfaces and the problem of uniqueness, Math. Zeit. 179 (1982) 151-168. 39. R. Osserman, A survey of minimal surfaces, vol. 1, Cambridge Univ. Press, New York (1989). 40. J. Perez, A rigidity theorem for periodic minimal surfaces, Comm. Anal. & Geom. 7 (1999) 95-104. 41. J. Perez & A. Ros, Some uniqueness and nonexistence theorems for embedded minimal surfaces, Math. Ann. 295(3) (1993) 513-525. 42. J. Perez & A. Ros, The space of properly embedded minimal surfaces with finite total curvature, Indiana Univ. Math. J. 45-1 (1996) 177-204. 43. J. T. Pitts & J. H. Rubinstein, Equivariant minimax and minimal surfaces in geometric three-manifolds, Bull. Amer. Math. Soc. (N.S.) 19 (1988) 303-309. 44. A. V. Pogorelov, On the stability of minimal surfaces, Doklady Akademii Nauk 260 (2) (1981) 293-295 MR 83b:49043. 45. B. Riemann, tiber die Fliiche vom kleinsten Inhalt bei gegebener Begrenzung, Abh. Konigl. d. Wiss. Gottingen, Mathern. Cl., 13 (1867) 3-52. 46. A. Ros, The Gauss map of minimal surfaces, preprint. 47. A. Ros, Compactness of space of properly embedded minimal surfaces with finite total curvature, Indiana Univ. Math. J. 44 (1995), 139-152. 48. A. Ros, Embedded minimal surfaces: forces, topology and symmetries, Calc. Var. 4 (1996) 469-496. 49. A. Ros, Peliculas de Jab6n, Fronteras de la Ciencia y la Tecnologia 14 (1997) 39-42. 50. R. Schoen, Uniqueness, Symmetry and embeddedness of minimal surfaces, J. Diff. Geom. 18 (1983) 701-809.

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J. Perez and A. Ros 51. R. Schoen, Estimates for stable minimal surfaces in three dimensonal manifolds, Volume 103 of Annals of Math. Studies, Princeton University Press (1983). 52. M. Shiffman, On surfaces of stationary area bounded by two circles, or convex curves, in parallel planes, Ann. of Math. 63 (1956) 77-90. 53. L. Simon, Lectures on geometric measure theory, Proc. of the Center for Mathematical Analysis, 3 Canberra, Australian National University (1983). 54. E. Toubiana, Un theoreme d'unicite de l'h!!.licoi"de, Ann. Inst. Fourier, Grenoble 38(4) (1988) 121-132. 55. M. Traizet, Gluing minimal surfaces with implicit function theorem, preprint. 56. M. Weber, The genus one Helicoid is embedded, Habilitationsschrift Universitat Bonn (1999). 57. M. Weber & M. Wolf, Teichmuller theory and handle addition for minimmal surfaces, preprint. 58. J. A. Wolf, Spaces of constant curvature, McGraw-Hill Series in Higher Mathematics (1967). 59. B. White, Complete surfaces of finite total curvature, J. Diff. Geom. 26 (1987) 315-326. 60. B. White, Curvature estimates and compactness theorems in 3manifolds for surfaces that are stationary for parametric elliptic functionals, Invent. Math. 88 (1987) 243-256.

Bryant Surfaces Harold Rosenberg Universite de Paris VII, Institut de Mathematique 2 Place de Jussieu, F-75251 Paris, France

Introduction In these lectures we will discuss the theory of surfaces in hyperbolic 3-space of mean curvature one. We call them Bryant surfaces. Robert Bryant showed how to parametrize these surfaces by meromorphic data and began the qualitative study of their geometry. Bryant surfaces have a meromorphic Gauss map and their intimate relation (they are cousins) to minimal surfaces in ]R3 has oriented their study. Many important properties and examples have been found by Umehara, Yamada, Rossman, Sa Earp, Toubiana and Zu-Huan Yu. We will present some of their results. My main goal is to present a theorem of Pascal Collin, Laurent Hauswirth and myself: a properly embedded Bryant surface in IHf3 of finite topology has finite total curvature and the Gauss map extends meromorphically to the conformal compactification. In fact a properly embedded Bryant annular end is asymptotic to a horosphere end or to a catenoid cousin end. Moreover if the end is part of a properly embedded Bryant surface which is not a horosphere, then the end is asymptotic to a catenoid cousin end. We will see this implies the only simply connected properly embedded Bryant surface is a horosphere, and the only such surface with exactly two annular ends is a catenoid cousin. We begin by a discussion of the theory of H-surfaces in the three simply connected space forms §3,]R3, and 1HI3 , and some problems are mentionned. Then Bryant's representation is described. We revue the theory of moving frames and indicate how this applies to Bryant surfaces. Examples are discussed. We then begin to prove our finite total curvature theorem.

1

Existence and unicity problems

The structure of constant mean curvature surfaces in the 3-dimensional simply connected space forms, is most difficult to understand in §3 and most transparent in IHf3. ]R3 is much closer to IHf3 in this sense; there are no non compact properly embedded examples in §3 (since §3 is compact) and these are the interesting surfaces we wish to understand in ]R3 and IHf3 . Consider the problems of existence and unicity (of embedded surfaces of constant mean curvature). Lawson constructed examples of oriented minimal

68

H. Rosenberg

surfaces in §3 of every genus [22]. It is easy to see that the totally geodesic 2sphere is the only (even immersed) minimal surface of genus zero. However we still do not know if the Clifford torus is the unique (up to ambient isometry) example of genus one. It is known that the space of minimal embeddings of fixed genus is compact [6], and two such examples are isotopic in §3 [23]. I know of no results of this nature concerning constant mean curvature (-:p 0) surfaces in §3. There are some unicity results known for minimal surfaces in §3 with boundary. Given a link r in §3, a minimal open book structure with binding r is a fibration of §3 - r by minimal leaves, each with boundary r. The simplest examples are r a geodesic and the fibration by minimal flat discs with boundary r, or r two dual geodesics and the fibration by flat annuli with boundary r. Then if M is an orientable embedded minimal surface (or immersed and of genus zero) with 1\:1 = r, the binding of a minimal open book structure, M equals one leaf of the fibration [11]. Thus if M is an embedded orientable minimal surface with 8M = r = a geodesic (or r= two dual geodesics) then M is a flat disc (a flat annulus). When r is two geodesics whose distance from each other is constant (but not necessary 7r /2) we do not know if one has unicity. Consider non zero constant mean curvature surfaces in IR3 (we'll call CMC surfaces); there has been much progress in our understanding of these surfaces. In the 50's, H.Hopf proved the only CMC immersed sphere in IR3 is the round sphere and Alexandrov proved the only closed embedded such surface is also a round sphere [14]. The techniques they used (a holomorphic quadratic differential associated to a CMC immersion, and Alexandrov reflection for embedded CMC's) have been essential tools of the subject since their discovery. In his 1970 paper, Lawson used the cousin correspondance (we will make this precise later) to construct doubly periodic CMC surfaces in IR3 from some minimal surfaces in §3. He constructed the minimal surfaces in §3 by solving Plateau problems for geodesic polygons and doing Schwarz reflection [22]. Hermann Karcher, using the cousin correspondance constructed many more CMC surfaces in IR3 [15]. He starts with minimal surfaces in §3, bounded by geodesic polygons, and considers their conjugate minimal surfaces in §3 (now bounded by planar lines of curvature geodesics) and takes the CMC cousin in IR3 of this conjugate surface in §3. The advantage of passing to the conjugate surface before taking the cousin, is that planar lines of curvature on a minimal surface in §3 are also planar lines of curvature (hence in symmetry planes) on the cousin. For example, the spherical helicoids (m, n -:p 0)

F(s, t) = cos s (

cosmt) sinmt 0

o

.

+ sm s

(

0

c~snt

smnt

)

Bryant Surfaces

69

are minimal surfaces in §3. The pieces 0 ::; S ::; 7r /2, 0 ::; t ::; 10, are simply connected and bounded by great circles. The corresponding cousins in IR3 are Delauney surfaces (embedded and immersed). Karcher has also applied this idea to construct embedded Bryant surfaces in JHf3 [19J. Start with a minimal surface M* in IR3, bounded by straight lines and simply connected. Its conjugate minimal surface Me in IR3 is then bounded by planar lines of symmetry. The cousin M of Me in JHf3 is then bounded by planar lines of symmetry as well. Now one extends M across its boundary by symmetry in the hyperbolic planes of the boundary curves. Using this technique he constructs symmetric n-noids, Schwarz P-type surfaces, and others.

2

The cousin relation

Let us now describe the cousin relation. Denote by M 3 (c) the simply connected space form of curvature c. The Gauss and Codazzi equations are the integrability conditions which guarantee that a simply connected Riemannian surface M, together with a smooth quadratic form II, can be realised isometrically in M 3 (c) so that II is the second fundamental form. More precisely, given (M, ds 2 ), M simply connected, together with a smooth field of symmetric transformations S of TxM --+ TxM, (IJ(X, Y) =< S(X), Y », satisfying

= det S + c (Gauss equation) (\lxIJ)(Y,Z) = (\lyIJ)(X,Z) (Codazzi equation)

Kds2

there exists an isometric immersion of Min M 3 (c) with II the second fundamental form. Here Kds2 is the Gauss (intrinsic) curvature of (M, ds 2 ) and S is the shape operator. The isometric embedding of (M, ds 2 ) is unique when one fixes a point and an oriented tangent plane at the point. Suppose now that M2 is a simply connected minimal surface in M 3 (c) with c = 1 or O. Let S be the shape operator of M and S± = S ± id.

Then traceS = 0, traceS± = ±2, and det S± = det S + 1 = Kext + 1 = C + 1. So (M2, ds 2 ) and S± satisfy the Gauss equation in M 3 (c - 1). The Codazzi equation also holds so (M2, ds 2 ) admits an isometric immersion in M 3 (c - 1) as a mean curvature one surface. This is the cousin of M in M 3 (c). Clearly one can reverse the process and go from a mean curvature one surface in M3(c - 1) to an isometric cousin in M3(c) that is minimal. So minimal surfaces in §3 have mean curvature one cousins in IR3 and mean Kds2 -

70

H. Rosenberg

curvature one cousins in §3 are the cousins of minimal surfaces in IR3. We will see that the + or - in S± produces quite a different cousin. Often one writes the cousin in terms of the second fundamental form, where I I± = I I ± ds 2 . One could also do this for the conjugate surface of a minimal surface in IR3 and more generally to the associate family M(O), o ~ 0 ~ IT /2, between M and its conjugate surface.

3

CMC's in IR3

In his book [14], H.Hopf posed the question: are there closed immersed CMC surfaces in IR3 other than the sphere? This became known as the Hopf conjecture and in 1986, H.Wente showed there are immersed CMC tori in IR3 [36]. Now there is a great deal known about the space of all such tori [5], [28]. Kapouleas discovered a desingularization technique to construct many new CMC surfaces in IR3, both compact and complete non compact such surfaces. He started with pieces of H-surfaces of revolution, the Delaunay surfaces, and glued them together to obtain approximate "balanced" CMC surfaces. Then he studies graphs in the normal bundle of the approximate surfaces and finds the CMC surfaces there, [15], [16]. This is a very flexible technique which has since been used to construct other types of surfaces of interest (we will discuss later). The theory of properly embedded (non compact) CMC surfaces really got off the ground with the paper of Meeks [25]. By a very beautiful geometric argument he proved any such annular end is cylindrically bounded. This implies, for example, that there are no properly embedded CMC's in IR3 of finite topology and just one end. Many interesting structure theorems (inspired by Meeks' paper) were obtained in [21]. They showed an annular end (properly embedded CMC) actually converges to an end of a Delauney surface. Also, they prove a finite topology CMC properly embedded surface with exactly two ends, is in fact a Delauney surface.

4

Some problems

I mention some old and new problems concerning unicity of CMC embedded surfaces in IR3 . 1. - Is a compact embedded CMC surface whose boundary is a circle, a spherical cap?

2. - Let M be a compact embedded CMC whose boundary is a convex planar curve in {X3 = O}. If M C {X3 ~ O}, does M have genus zero, i.e., is M topologically a disk? 3. - Let C 1 and C2 be planar convex curves in disjoint parallel planes. Is there a compact annulus with boundary C 1 UC2 and constant mean curvature?

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71

4. - Let M be a properly embedded CMC surface with compact boundary. If M is cylindrically bounded, is the genus of M finite? 5. - Can one desingularize two touching hemispheres (of the same curvature) to obtain a CMC surface of non zero genus? More precisely, let M be the connected union of the two hemispheres (of H = 1) bounded by the two circles Co = {xi +X~ = 1,x3 = o} and C1 = Co + (0,0,2). Does there exist a CMC surface N, close to M, of genus ¥- 0, and whose boundary is convex planar curves (in the planes X3 = 0, X3 = 2) close to Co and C1 ? 6. - Let r c {X3 = O} be a strictly convex curve, and D the planar disc bounded by r. Fix V > and let M be a surface with M = r such that MUD bounds a domain Q whose volume is V. Minimizing the area of such M, geometric measure theory techniques show there is an embedded such M that minimizes area among surfaces bounding (with D) a volume V. Note such an M by M(V); M(V) has constant mean curvature. Clearly M(O) = D and for V > 0, V near 0, M(V) is a graph over D (in X3 > 0, say). The problem is to understand the behavior of M(V) as V --7 00. It seems likely that the only r for which the M (V) will stay in the half-space X3 > 0, is the round circle. For a long thin ellipse, the M (V) appear to sink rather rapidly (computer experiments done by David Hoffman [12]) into the half-space X3 < 0. And this takes place near the part of the ellipse with small curvature.

°

Now if M(V) does dip down into X3 < 0, perhaps the two descending parts would touch for a certain V; cf. figure 1. If they do touch then M (V) acquires a handle there and M (V) would be topologically a torus with a disk removed. We know no example of an embedded compact H-surface of genus greater than zero, whose boundary is a convex planar curve.

a cross section before touching figure 1

72

H. Rosenberg

7. - Let M be a properly embedded H-surface contained in a slab of 1R3 ;i.e., M is between two parallel planes. Does M have a symmetry plane inside the slab? Using Alexandrov reflection it is not too hard to show this is true if M has finite topology so the question concerns infinite topology surfaces. When M has bounded curvature one can begin to do Alexandrov reflection coming down from the top with horizontal planes (we fix the slab that contains M to the horizontal). Then either there is a symmetry plane or the first accident occurs at a plane P where the symmetry of the part of M above P (M+ say) is above M- but asymptotic to M- at infinity. This "tangence" at infinity must in fact take place along M n P, and the tangent plane to M becomes vertical when going to infinity along a branch of MnP. There is no general maximun principle at 00 along the boundary. Ronaldo Freire, in his thesis, establishes a maximum principle at infinity for H-surfaces when the boundaries are at a strictly positive distance. 8. - Bill Meeks asks the question: If M is a properly embedded H-surface in a half space of 1R3 (X3 > 0 say), does M have a horizontal plane of symmetry? In particular, M would then be in a slab.

5

H-surfaces in

JH[3

For H > 1, the theory of properly embedded H-surfaces in JHf3 is very much akin to the theory of H-surfaces in 1R3 , and some results are easier to come by (not only for H > 1). I believe this to be for two reasons: there are more symmetry planes to work with and the asymptotic boundary of JHf3 is so much richer. Here are some examples -Meeks' theorem on annular ends (H

> 1) is true.

-An H-surface of finite topology and exactly two ends is a Delauney surface

(H > 1), [20]. -Properly embedded annular ends (H [20].

> 1) converge to Delauney ends

-A compact embedded H-surface, with boundary a circle is a spherical cap (H::; 1) [27]. This is even true for immersed such M [3]. -If M is a properly embedded H-surface and the asymptotic boundary, 000 M, is contained in a circle of 5 00 then M is invariant by symmetry through the (hyperbolic) plane containing the circle and the part of M on one side of this plane is a graph over the plane (a part of the plane) with respect to the geodesics orthogonal to the plane [24]. In particular if oooM is one point then M is a horosphere (this was first proved by Do Carmo and Lawson), and if 8 00 M is two points, then M is invariant by rotation about the geodesic joining the two points at infinity.

Bryant Surfaces

6

73

Properly embedded minimal surfaces in JR3

We now know many examples of properly embedded minimal surfaces in ]R3, both of finite and infinite topology. Periodic minimal surfaces have been known for hundreds of years; Scherk's singly periodic surface is perhaps the first infinite genus example discovered. Scherk found his surface by solving the minimal surface equation over planar domains (separation of variables). Today we construct periodic examples using the Weierstrass representation or by solving the Plateau problem for a suitable polygonal Jordan curve and then doing Schwarz reflection along all the line segment boundaries that present themselves by symmetry. The reader may consult [18] or [30] for a discussion of these examples. Now we know how to construct infinite genus minimal surfaces by "desingularization" . The first to do this was Martin Traizet who desingularised the intersection of vertical families of planes [31]. Near each line of intersection, the desingularized surface looks like the singly periodic Scherk surface. Subsequently, Kapouleas desingularized certain families of catenoids, intersecting in circles; the desingularized surface looks like a "Scherk collar" near each circle. Recently Kapouleas has announced that he can desingularize more general transverse intersections. This produces many non periodic infinite genus examples. This can even be done keeping the curvature (Gaussian) bounded [29]. Until 1982, the only known minimal surfaces (we will always suppose properly embedded in the sequel) in ]R3 0 f finite topology (i.e. finite genus and a finite number of ends) were the helicoid, plane and catenoid. then in 1982, C.Costa discovered an example of a minimal torus with three embedded ends and Hoffman and Meeks proved this surface is embedded. They subsequently constructed many new examples of finite topology [13]. The outstanding achievement of the past decade in this subject was Pascal Collin's solution of the generalized Nietsche conjecture [4]. From this we know that any minimal surface of finite topology and at least two ends in ]R3 is of finite total curvature; all the ends are asymptotic to planar or catenoid ends. This implies the only annular such surface is the catenoid (or more generally, the only finite topology minimal surface in ]R3 with two ends is a catenoid). The helicoid has one end, trivial topology, and infinite total curvature, and to this day, we still do not know if the helicoid is unique. Is the helicoid the unique (non planar) properly embedded simply connected minimal surface in ]R3? Since the helicoid was discovered in 1776, this is indeed an old question! Another question of this nature: does a properly embedded minimal surface whose curvature tends to zero (as p E M tends to infinity) have finite total curvature?

74

7

H. Rosenberg

Bryants' representation

We will now describe the manner by which Bryant parametrized H = 1 surfaces in IHf3 by meromorphic data [2]. There are three standard models for IHf3 and one other which we shall use. Let L4 denote the Minkowski 4-space, which is ]R4 together with the quadratic form - x6 + xi + x~ + x~, and the orientation dxo 1\ dx 1 1\ dX2 1\ dX3 . Then IHf3 is the submanifold

{v E L4/ < v,v >= -1 and xo(v) > O}, and the Minkowski quadratic form restricts to a Riemannian metric on IHf3 . The ideal boundary Soo of IHf3 , in this model, is the space of null lines, passing through the vectors v in the light cone « v,v >= 0) with xo(v) > O. The geodesics of IHf3 are the 2-planes of L4 , through the origin, that intersect IHf3. Each geodesic converges to two points of Soo, defined by the two rays where the 2-planes meets the light cone. The ball model B of IHf3 is obtained by projection of IHf3 into the plane Xo = 0, from the point Xo = -1, Xi = 0, i = 1,2,3. The image of IHf3 is the ball B = {xi

+ x~ + x~ <

1}, together with the metric ds 2 = (1

~d~2xn'

with dx 2 the euclidean metric. The third model is the upper half space model { (Xl, X2, X3) / X3 > O} together with the metric ~. The last model is obtained X3 by identifying L4 with the hermitian symmetric 2 x 2 matrices by identifying (xo, Xl, X2, X3) with

Then IHf3 is the set of such matrices v with det v = 1. The action of SL(2, q on these hermitian matrices defined by

v --t gvg*, g*

=t

g,

preserves the inner product (since det v = - < v,v » and leaves IHf3 invariant. PSL(2, q is the group of isometries of IHf3. Notice that the map SL(2, q --t Herm(2 x 2), F --t FF* takes its values in IHf3. Now we can describe the Bryant representation. For

(~~)

E SL(2, q

a matrix of holomorphic functions of a variable z, we let 9 = - ~~ and w = AdC - CdA. Theorem 7.1. (Bryant [2]). Let M be a Riemann surface and F : M --t SL(2, q a conformal immersion such that det(F-1dF) = O. Let f : M --t IHf3 be the map F F*. Then f is a mean curvature one immersion of M in JHl3 • Conversely, given an immersion f : M --t JHl3 of mean curvature one, there

Bryant Surfaces

75

exists a holomorphic lifting of f to the universal cover F : M ~ Sl(2, C), such that F is a null-immersion, i.e. det(F-1dF) = 0, and f = FF* (on simply connected domains of M, F is determined up to right multiplication by a constant in SU(2)). ~

Remark 7.2. Consider a null immersion F : M locally write

Sl(2, C). We can always

F-1dF = (g _g2) w 1 -g ,

(7.1)

where 9 is meromorphic and w a holomorphic one form. For example, write:

A(Z) B(Z)) F(z) = ( C(z) D(z) E SL(2, C) where z is a local conformal coordinate and A, B, C, Dare holomorphic. Then 9 = - ~~, w = AdC - CdA, solves (7.1). Since F is holomorphic, the forms w, g2w are also. F is an immersion so F-1dF is never zero. It follows that the set of poles of 9 coincide with the zeros of wand a pole of 9 of order k is a zero of w of order 2k. We call (g, w) a Weierstrass pair associated to F. The pair is defined up to multiplication of F on the right by a constant in SU(2). So if F is replaced by Fh, h in SU(2), then F-1dF becomes h* F-1dFh, and the new ag - b (b 9 = bg + a' w = 9

The formulas for

Cg, w)

_)2

+a

Cg,w)

= (a-bab)

are

w.

show that the holomorphic 2-form

Q =wdg is globally defined. In fact, the metric on the immersion in terms of (g,w) by ds 2 = (1

+ gg)2 WW

= (1

f :M

~

IHf3 is given

+ Ig1 2) 2 1w12 ,

and Q is the complexification of the second fundamental form II of M, which is -Q - Q + ds 2 . These facts will become clear later. Now the (simply connected) surface M together with a Weierstrass pair (g,w) and metric ds = Iwl (1 + IgI 2 ), admits an isometric minimal immersion in ]R3 via the Weierstrass representation:

76

H. Rosenberg

and the second fundamental form of this immersion in ]R3 is given by IIR,3 = -2Re(wdg) = -2ReQj here M is oriented by the Gauss map g. Since the second fundamental form of M in IHf3 is - 2ReQ + ds 2 , M in IHr is the cousin of Min ]R3. Given a meromorphic map h on a Riemann surface, the Schwarzian of h is a meromorphic quadratic differential which is given in a local conformal coordinate z, by S(h) = Sz(h)dz 2, where

Sz(h) =

(h!h" ) - '12 (hll) h! I

2

Then Umehara-Yamada observed that

S(g) - S(G)

= 2Q

where G is the hyperbolic Gauss map of M. Since Q is holomorphic on M, the (eventual) poles of S(g) and S(G) must cancel. A straightforward calculation of Sz (h), at a pole or zero of h of order k, shows that

At z E M, the coefficients of

z12

of Sz(g) and Sz(G) must be the same, so

k(g)2

= k(G)2.

In particular if G has a branch point at z (i.e. kz(G) = b + 1, b > 0 the branching order of G at z) then so does g (or ~) and of the same branching order. Thus the contact of the minimal surface M in]R3 (coming from (g, w)) with it's tangent plane at z, is the same as the contact of the cousin in IHf3 with its tangent horosphere. We will now indicate some of the steps in Bryants' representation. The geometry in IHf3 here is best described using adapted moving frames. We begin with a brief section describing the general theory of moving frames.

8

Moving frames

In IHf3, calculations are facilited by using moving frames. Before presenting Bryant's representation of H = 1 surfaces in IHf3 , I summarize here the technique of moving frames for a submanifold Mm of a Riemannian manifold Nn. Let el, ... , en be an orthonormal frame of N, V' the Riemannian connexion of N and ri~ the Christoffel symbols:

Bryant Surfaces

R(X,Y)Z

77

= V'xV'yZ - V'yV'x Z - V'[X,YJZ,

the curvature. Let WI, ... , wn be the dual one forms of eI, ... , en; wi(ej) = n x n matrix of) one forms wI by

J). Define (an

V'ei = wI(.)ej, i.e., V'ekei = wl(ek)ej,or w{(ek) =< V'ekei,ej >, (so w{ = -w}). The (connexion) forms (w{) satisfy the structure equations: dw i dw Ji

+ Wi /\ w j = 0 + wJki /\ wJk = [liJ'

where [lJ are the curvature 2-forms: iIi

k

I

[lj = 2RjklW /\ w , (we remark that the equations dw i + w) /\ w j

= 0, wI = -wJ, characterize the

wI)· Taking the exterior derivative of the structure equations one obtains the first and second Bianchi identities: wi /\ [lij = 0, i.e., [l/\ w = 0, and d[l + w /\ [l - [l/\ w = o. Or in terms of R's:

(for the first equation, apply the 3-form

to (ep,eq,e r )). Now we consider a submanifold Mm C Nn and we work with adapted frames (eA) of N, A = 1, ... , n, where the first m vectors ei, i = 1, ... , m are assumed tangent to M and (eo), a = m + 1, ... , n denote the remaining vectors. The connexion forms (w~) satisfy

dw A + w~ /\ wB = 0

w~

+w!J = O.

Restricting (w~) to M gives the Riemannian connexions forms of M:

78

H. Rosenberg

dw A = -w~ /\ wB restricted to M gives: dw i = -w} /\ wj , i,j = 1, ... ,m, since wO: = 0 on M. Since w} = -wI, the matrix (w}) is the Riemannian connexion form of the induced metric on M. Now wO: = 0 on M so 0 = dwO: = -wAO: /\ w A = - ",n /\ wi , so by Ut= 1 w~ t Cartan's lemma, there exist functions (hij) such that wi0: --

' "' ~

hO:ijw. j

j

The hij are the coefficients of the second fundamental form II in the direction eo:: m'

n

~

\7eo: = Lj=l w~ej + L~=m+l wo:e~ = w~ej + w~e~.

j so < \7e a, -dx >= _Wia wi = -w~wi hO:·wiw 'l. 1 , J . Let (\7 X~)T = -A~X for X tangent to M, ~ orthogonal to M, and T the tangent component. Then (\7 e;eo:)T = (w~(ei)eA)T = w~(ei)ej = -Ae" (ei). So A e" (ei) = -w~(ei)ej = wj(ei)ej = hjkwk(ei)ej = hjiej. Next we derive the Gauss, Codazzi, and Ricci equations. From dwO: + w~ /\ wB = 0, restricted to M..L: Define dx

= wie'

1.,

= dwO: +w$ /\w~ = 0

dwO: +w13 /\w B

(since Wi

= 0 on

M..L).

Also w3 = -w~ so (w3) is the connexion matrix of M..L.

and restricting indices: i

dw j

+ wki /\ Wjk + Wo:i /\ Wj

0:

~

= Jlij .

Comparing this with the 2nd structure equation of M: dw} +wl /\wj = Jlij , and using

Wo:i

/\

Wj0: -_ - hO:ikW k /\ hO:jlW I -- - hO:ik hO:jlW k /\ WI ,

n

we obtain ij - Jlij = -hikhjlW k /\ wi. Evaluate on e r , e s to obtain the Gauss equation:

R ijrs = -hikhjl(8~8~ - 8:8~) = h?shjr - h?rhjs , e s ) = Rjrs = R ijrs , and (w k /\ wi) (X, Y) = wk(X)wl(y) -

RijrS -

(recall Jlij(e r wk(Y)wl(X)).

Bryant Surfaces

79

Working (a little harder) one obtains the Codazzi equation:

HG:irs = (\1 e r hG:) (e s , ei) - (\1 e, hG:) (en ei) +h/3(e s ,ej) < \1te/3,eG: > -h/3(er ,ei) < \1-;' e/3, eG: >, and the Ricci equation: ~ R rSG:/3 - R.L rSG:/3 -- hG:is h/3ir

-

hG:ir h/3is'

When the codimension of M is one, the normal connexion is trivial so the Codazzi equation becomes:

HG:irs = (\1 er hG:) (es,ei) - (\1 e ,hG:) (er,ei). Also if N has constant curvature, then H(X, Y)Z is tangent to M when X, Y, Z are tangent to M. Finally Codazzi (in this case) becomes:

\1xII(Y,Z) = \1yII(X,Z).

9

The structure equation of H3 and Boo

Denote by F the six manifold of bases (eo, el, e2, e3) of 1I} satisfying eo /\ el /\ e2 /\ e3

>0

xO(eo) > 0

< eG:,e/3 >= {

-I if ex, (3 =0 0 if ex::fi (3 1 if ex = (3 = 1,2 or 3

Then eG: : F ---+ 1I} is a vector valued function so there exist unique I-forms on F, {wg! ex, (3 = 0,1,2, 3} such that

deG: = w~e/3 Let the indices 1 :S i, j, k :S 3 and write Wi for tion of < ., . > yields

wb. Then a simple applica-

deo = wiei dei = wieo + wI ej 0= w} + wf. Taking the derivative gives the structure equations

+ wiJ /\ w j = 0 dw} + w1/\ wj + Wi /\ wj dw i

= O.

Clearly eo : F ---+ JHf3 is a smooth submersion and el, e2, e3 E Teo (JHf3 ), so F is the oriented orthonormal frame bundle of JHf3 and if ds 2 denotes the metric on JHl3 then

80

H. Rosenberg

e~(ds2)

=< deo, deo >= (W I )2 + (W 2)2 + (w 3)2.

The map eo + e3 takes its values in the positive light cone N3 and d( eo + e3) = w3(eo + e3) + (WI + w§)el + (w 2 + w~)e2' so < d(eo + e3), d(eo + e3) >= (wI + w§)2 + (w 2 + W~)2. Let da 2 be the induced form on N 3, so the above quantity is (eO+e3)* (da 2). We have Soo = N 3/IR+ = the rays in the positive light cone and if [eo + e3] denotes the ray through eo + e3 then [eo + e3] : F ---+ Soo. Soo is oriented by (w 2 + w~) A (WI + w§). Consider the basis of lI} eo

=

(°110)

,el

=

(01°1)

,e2 =

(0 °i) -i

,e3 =

(1 -10) ' 0

and define ea(g) = g.e a = geag*, for 9 E SL(2, C). This identifies PSL(2, C) with F : 9 ---+ (e a (g)). The forms on F, Wi, w), pull back to the left invariant one forms and

10

Surfaces in IHI3 and the structure equations of adapted frames

Let f : M2 ---+ JHf3 be a smooth immersion, M an oriented surface. We now consider the canonical forms of F when they are adapted to M and derive the structure equations of M. Let F(f) denote the adapted frames of lI} to M: (m,eo,el,e2,e3), with m E M, eo = f (m), el ,e2 an oriented basis of Teo (M) (el A e2 the orientation), and e3 normal to M in JHf3 . The induced metric on M is ds} =< df, df > and since w3 =< e3, df >= 0 on F(f) we haveds} = (W I )2+(W 2)2, and the area form of Mis dA f = WI Aw 2. We have dw 3 = 0 = -wr AWl -w~ AW2 on F(f) so by Cartan's lemma we know there exist smooth functions (hij ), i, j = 1,2, hij = hji' such that

The quadratic form II = hll (w l )2

+ 2hl2W l W2 + h22(w 2)2

is the second fundamental form of M in JHf3 . Define w = WI + iw 2 (this w is not the w of a Weierstrass pair), so that ds} = w.w, dAf = ~w A wand dw = A w,

-iwr

dwr =

-w~ A

wr +

WI

AW2 = (1

+ hi2

- hllh22)

WI

A w2 .

Bryant Surfaces

81

Thus -K = 1 + hr2 - hl1h22' where K is the intrinsic curvature of M (this is the Gauss equation). The mean curvature is H

1

= 2" (hl1 + h22 ) .

At eo = f(m), the geodesic ofIHfl with tangent e3 meets Soo at [eO+e3] (and at [eo - e3] in the other sense). The map G : M -+ Soo, G(m) = [eo + e3](m) is called the (hyperbolic) Gauss map of M. A straightforward calculation then yields Proposition 10.3. [2} G : M -+ Soo is conformal iff f is either totally umbilic (in which case G reverses orientation) or f satisfies H == 1 (in which

case G preserves orientation). When H == 1 and f is totally umbilic, then f(M) is contained in a horosphere, G is constant, and its value is the asymptotic value of the horosphere. Thus G is analogous to the Gauss map of minimal surfaces in ffi.3. On a minimal surface M C ffi.3, with K the Gauss curvature, - K ds 2 is a metric of Gauss curvature 1 on M (where K :I 0) and it is the pull back of the constant curvature 1 metric on the unit sphere, via the Gauss map 9 of M. Conversely (by a theorem of Frobenius), if ds 2 is a metric on M of Gauss curvature 1 then there exists an isometric immersion of (simply connected pieces of) Minto ffi.3 as a minimal surface. We have a similar situation for an immersion f : M -+ IHfl with H == 1. Let "1 = (wI + w~) - i(w 2 + w~), so "1 = ((1 - h l1 ) + ih l2 ) w ~TJAi] = -K~WAW.

The structure equations give

dw = -iwr Aw dTJ = iWf A "1 dWI2 = - KiZW 1\A W-

i = zTJ

A-

1\

"1.

Thus daJ = TJ.i] = -Kw.w = -KdsJ has Gauss curvature 1. As before the Frobenius theorem yields that if - K ds 2 is a metric on M of Gauss curvature one, then (locally) M can be isometrically immersed in IHfl with H == 1. At this point the reader can refer to Bryants' paper for the proof of Theorem 1; we have presented the necessary ideas to understand the proof.

11

Constructing explicit examples of Bryant surfaces starting with a minimal surface in IR3

In his paper [2] Bryant calculated several cousins of minimal surfaces in ffi.3; some catenoids and an Enneper surface. Umehara and Yamada gave techniques to construct examples in terms of certain differential equations and

82

H. Rosenberg

they derived many examples [32]. It turns out that the cousin of a catenoid depends on which catenoid one chooses. Not only will the normal orientation of the catenoid change the geometry of the cousin but homothety will produce completely different cousins (they are not always rotational). Let us see how this works with the catenoids in R3. I am grateful to Pascal Collin for finding the symmetric solutions of the differential equations and the computer pictures (the solutions already were calculated in the literature but not as symmetric and sometimes erroneously). We will obtain the surfaces in the upper half space model of 1Hf3. Since our data will come from 5L(2, q we must choose an orientation preserving isometry from 1Hf3 in II} to the upper half space model. The reader should be warned that different workers in the field choose different isometries, so the formulae are not necessarily the same! From 1Hf3 c II} to the unit ball B = {xi + x~ + x~ < I}, there is a very natural projection by taking the line segment from x = (XO,Xl,X2,X3) E 1Hf3 to (-1,0,0,0), and seeing where it intersects B, (at y):

::::;.y-(o~~~) -

, Xo

+ 1 ' Xo + 1 ' Xo + 1

or Yi = x:.il' i=1,2,3 (cf. figure 2). Now we want an orientation preserving isometry from B to the upper-half space model {(u, v, w)/w > O}. First do an inversion in the sphere of radius v'2 centered at (0,0,1):

2(p-(0,0,1)) (0,0,1)112

p -+ lip -

+ (0,0,1).

This is orientation reversing and takes B to the lower half-space {w < O}. So compose this map with (u,v,w) -+ (u,v,-w), to obtain the orientation preserving isometry:

Since

one obtains

Bryant Surfaces

(u,v,w)

One can solve for

Xi

=(

Xl

Xo - X3

in terms of F

1) .

, Xo X2- X3 , Xo -

= (~~) +

iX2) ( xo +:C3 Xl Xl - ZX2 Xo - X3

83

X3

E 5L(2, C) from the relation

= FF*

to obtain

-1

figure 2

2xo

= IAI2 + IBI2 + IGI2 + IDI2

2X3

=

IAI2 + IBI2 - IGI2 - IDI2 + iX2 = AG + BD Xl - iX2 = Gll + D B 2XI = AG + Gll + BD + DB 2X2 = i (G A + D B - AG - Gll) Xl

Finally we obtain (u, v, w) in terms of A, B, G, D:

.

84

H. Rosenberg

u +iv

AC +BD

= ICl 2+ IDI2 1

w

101 2+ IDI 2'

Now we can look at the catenoid cousins. The catenoids in strass data

m.3

have Weier-

g(z) = eZ,w(z) = )..e-zdz, (usually one writes g(z) = z,w(z) = )..~ on C*, but we lifted this pair to is the inward pointing normal. Now the metric of these catenoids is ds = Iwl(1+lgI2) = 2)..cosh(x)ldzl,z = x + iy, and the second fundamental form is given by

°

II

= -2Re(Q) = -2Re()"dz 2 ),

so for v = a + ib a tangent vector,

II(v, v) = -2)..(a 2

-

b2 ).

The cousin in IHf3 has second fundamental form II = - 2Re (Q)

II(v, v) = -2)..(a 2

-

b2 )

+ ds 2 ,

so

+ 4)..2 cosh2 (x)(a 2 + b2 ).

Consider f, the image of the waist circle, x = 0, v = (0,1), so ll(v, v) = 2),,(1 + 2),,). The waist circle is a geodesic on the catenoid and a line of curvature on the cousin, so f is a geodesic line of curvature. Let A denote the shape operator of the cousin, so A(v) = k1v, kl the normal curvature of f. 2),,(1 + 2),,) 1 Since II(v, v) =< Av, v >= k 1 1lvl1 2= kl 4)..2 , we have kl = 4)..2 = 1+ 2),,' For)" > 0, we will see that f is a circle (the cousin is a surface of revolution) and f converges to a point as ).. -+ (k 1 is the curvature of fin IHf3 since the geodesic curvature is zero). For ).. < 0, we are taking the cousin of the catenoid with the outer orientation. Now kl = -1 means f is a horocycle, and since kl = 1 + this happens when).. = When kl = (which happens when).. = -~), f is a geodesic of IHf3. We will see these examples when we explicitly calculate the cousins. For)" > 0, the cousins are embedded rotational surfaces. For - < ).. < 0, the cousins are rotational non embedded surfaces. For ).. = the cousin is ruled by horocycles and not embedded. the cousin is a cone on a spiral in the horosphere X3 1, For ).. < containing a vertical geodesic for A = - ~.

°

°

2\.'

i.

i

i, i,

Bryant Surfaces

85

To find the cousins we solve

that is

(AB) (e-z1 -e

dF = >. CD

Z

-1

)

dz.

So we get a system of equations for A, B, C, D:

A' = (I) B' = C' = D' =

>'A + >.e- Z B - >.Ae z - >'B >.C + >.e- z D ->.ezC - >'D.

To solve this one proceeds as in [32J (They show how to do it in general).

A"

= >.A' -

>.Be- z + >.e- z (->.Ae z - >.B) + l)>'Be- z + l)(A' - >.A)

= >'A' - >.2 A - (>. = >'A' - >.2 A - (>.

so A satisfies the equation

A" + A' - >'A

= 0,

and C also satisfies the last equation. Repeat this process for B:

B" = ->.B' - >.Ae z - >.e z (>.A + >.e- z B) = ->.B' - >.2 B - (>. + l)>.Ae z = ->.B' - >.2 B + (>. + l)(B' + >.B) so

B" - B' - >'B = O. Then to find F (Umehara and Yamada [32]) solve the system

(II) A" + A' - >'A = 0 (same for C) B" - B' - >'B = 0 (same for D). the equations {

"12

+ "I -

62

-

>. = 0

6 - >. = 0

have roots "II = -~ + a, "12 = -~ - a, 61 = ~ + a, 62 = ~ - a, where i + >.. Suppose>. iLook for solutions A = ae'YIZ, B = be 01Z • The system (I) becomes

a2 =

-i.

86

H. Rosenberg

¢} {

t)

(a - a = Aa + Ab (a + 2") b = - Aa - Ab

(a - ~) (a +~)

A= so

= 1161 = 1262

{ a=(a+~)a+(a+~)b b=(~-a)a+(~-a)b

¢}

(a-~)a+ (a+~)b=O

and

A= (I II)

B =

C= D =

r (a +~) e(n-!)z r (1 - a) e(n+!)z 2 ( s (~ - a) e -!-n)z s (~+ a) e(!-nV

One can also replace a by -a in (III) to obtain solutions. Since F E SL(2, q,

1 = AD - BC = rs [(~ = 2rsa

+ a)2 - (~ - a)2]

We can now solve for (u, v, w). We take r = s, and first consider the case and a i- ~. the reader can then verify

a real, a > 0,

u+iv = w

This is a surface of revolution about the w axis (x fixed, y varies, are circles centered at (0, 0, w)). The waist circle on the catenoid in ]R3 is {x = O} and this is the circle

Bryant Surfaces

87

To understand the behavior as a -+ ~ (>. -+ 0) we do the homothety from (0,0,0) ( a hyperbolic isometry) that brings the waist circle to height one: multiply (u,v,w) by +( 2 ) la. Then the waist circle of the catenoid goes to

U

(u

+ iv)(O,y)

=

Ci- - ( 2 ) e2iya

= 1.

X3(0,y)

a

This is a circle of zero radius for a = ~ (>. -+ 0) and infinite radius for a=O(>.=-i)· To understand the limits, let x = In(t) + x', x' bounded, y = y', a = ~ + t. Then

°

When t -+ , e2tx -+ 1, te- X -+ e- x ' , and U

+ iv ---+ -

-x

,

e

e- 2x ' + 1 1 ---+ e- 2x ' + 1 .

w

e'Y

The reader can check that this is part of the horosphere which is the euclidean sphere of radius ~ centered at (0, 0, ~). Now make the change of variables x = -In(t) + x', x' bounded, y = y'. As t -+ , e2tx -+ 1, te X -+ eX' and

°

u

+ iv ---+ -ex' eiy '

w

---+ 1.

This is an annulus in the plane X3 = 1. So we have an idea of how the family of catenoid cousins is converging to the union of the two horospheres tangent at (0,0,1) as >. -+ 0, >. > 0: .Imaginarya '(3 (3 rea, I (3 > ,s - V21J 1 . (2 rsa - (-2i)'(3 Let a -- z, an d r - -zs 2/3 Z -- 1) . Then (III) yields

° -

88

H. Rosenberg

Write this as

u + iv

= Ql(x)e- 2/3y

w

= Q2(x)e- 2/3y

This is a cone from (0,0,0) on the trace curve in w = 1 (which spirals). We saw that the waist circle went to an infinite radius circle (on the normalized solution) when A -+ - ~, A real. Let us examine this more closely. The normalised solution is

u+iv =

(~

w

+ ex 2) (~ _ ex 2) (eX + e- X) e2a (x+iy) ex [( ~ - ex) 2 e-x + (~ + ex) 2 ex] 2 (~ + ex 2) e 2xa 1

Do a horizontal translation by

u'

4-;''''

2

,so that (u'

+ iv')(O)

= O. Then

+ iv' =

w

Fix z = x u

,

+ iy .,

and let

ex -+

0:

1 [~(ex+e-X)(2(X+iy))-(eX-e-X)]

+ w -----+ -

4

+ iy

~(ex+e-x)

eX - e- x --22 eX + e- X x

v' This limit surface is ruled by horocycles in the horospheres w=constant (the lines x fixed, y varies). The following figures are the catenoid cousins in the upper half space model for various values of A.

Bryant Surfaces

>'=1

>. = 0.3

>. = 0.05

>. = om

89

90

H. Rosenberg

A = -0,01

&-----&.6f"F?it A = -0.02

A = -0.05

A = -0.24

A = -0.25

Bryant Surfaces

A = -0.27

A = -0.5

A = -1.5

91

92

12

H. Rosenberg

Properly embedded Bryant annular ends

Let E C JHl3 be a properly embedded Bryant annular end. We will now discuss the proof that E has finite total curvature and is regular [7]. We subsequently assume E is not contained in a horosphere. Let 800 E denote the asymptotic boundary of E in 8 00 , the sphere at infinity. We first consider the case when E is not dense at infinity (Le., 800 E =F 8 00 ), and then we will show E can not be dense at infinity. An important tool is the tangent horosphere of E at a point q in the interior of E. This is the unique horosphere H(q), tangent to E at q, whose mean curvature vector at q coincides with the mean curvature vector of E at q. The point at infinity of H(q) is the hyperbolic Gauss map image G(q) of q E E. As we pointed out in section 8, the contact of E with H(q) at q, is the same as the contact of the cousin minimal surface with its tangent plane at the "cousin point" of q. Thus the intersection of E and H(q) is an analytic curve near q with a singularity at q where there are 2k + 2 smooth branches meeting at equal angles. The integer k is at least one; k - 1 is the order of q as a branch point of G and k is the order of contact of the cousin minimal surface with its tangent plane. We introduce a mean convex side W of E. Let E be a compact orient able embedded surface with 8E = 8E and EnintE = 0. Then EUE separates JHl3 into two connected components and we let W be the component along which E is mean convex. The constructions we will make take place far from 8E so the choice of E is not important. When E is part of a properly embedded Bryant surface M, then W is taken to be the mean convex side of M. We denote by H(q)+ the mean convex component of JHl3 - H(q) (the inside of H(q)) and let H(q)- denote the other component (the outside of H(q)). Here is the fundamental theorem concerning H(q)

n E.

Theorem 12.4. (the horosphere theorem) Let q E intE and assume En H(q) is compact and disjoint from 8E. Then if 8E C H(q)+, every divergent path in JHl3, starting at x E 8E, must intersect E U H(q) at a point other than x. If 8E C H(q)-, then every divergent path starting at x E 8E and staying in W, must intersect H(q). The two possibilities are presented in figures three and four. When the connected component E(q) of q in E U H(q) is compact, disjoint from 8E, and E is transverse to H(q) along E(q) - {q}, then E(q) is a figure eight, the union of two Jordan curves Gl , G2 meeting at q, and k = l. The proof of the horosphere theorem is based on an analysis of the connected components of E - E(q). The fact that the genus of E is zero is fundamental here. The starting point is the observation that there is no compact component F of E - H(q), outside of H(q), whose boundary is contained

Bryant Surfaces

H(qr

/

G(q)

figure 3

H(q)

H(qr

G(q)

figure 4

93

94

H. Rosenberg

in H(q). IHr is foliated by horospheres parallel to H(q) ( a constant distance from H(q)) and if such a component F would touch a leaf of this foliation at a point y E F, furthest from H(q). Then the maximum principle would imply F equals the leaf through y, i.e. E is contained in a horosphere. Similarly, if such a component F of E - H(q) were inside H(q) (and 8F C H(q)) then the above foliation implies F together with a compact domain D C H(q), form the boundary of a compact region Q of IHr along which F is mean convex. These properties of the components of E - H(q) and genus of E equals zero, enables one to prove the Horosphere Theorem. The reader should keep in mind the two figures. Now we indicate the ideas of several theorems leading towards our main annular end theorem. Theorem 12.5. Let E be a properly embedded Bryant annular end. If 800 E -I 5 00 (i. e., E is not dense at infinity) then E is conformally a punctured disk and G extends meromorphically to the puncture (i.e., E is a regular end). Proof. [Idea of the proof.] We will now work in the upper half-space model of IHr with 5 00 = {X3 = O} U {oo}. Since the asymptotic boundary of E is closed and not 5 00 , we can assume E C B = {xi + X~ + x~ < R2, X3 > O}.

First we will show that G is bounded on some subend of E. If not, then for some qn E E, diverging on E, we would have tG(qn)t -+ 00. Since 8E is compact, we have x31aE 2 J > 0 for some 6. Choose n sufficiently large so that H(qn) n B is below X3 = 6. This is possible since X3 (qn) -+ OJ cf figure 5 . However, 8E C H(qn)+ and we can find a path from 8E to G(qn) which does not intersect E U H(qn) except at its endpoint (choose a path from a point of 8E to a point of 8B, not meeting E then choose n big enough so that H(qn) is below this path, and then continue to G(qn)). This contradicts the horosphere theorem so G is bounded. Now to complete the proof of theorem 12.5, it suffices to show E is conformally a punctured disk (since G is bounded so could not have an essential singularity at the puncture). To obtain the conformal type one uses the dual surface E# introduced bX Umehara and Yamada [33). This is the immersion of the universal cover E into IHr whose lifting to 5L(2, q is F# = F- 1 , where F : E ~ 5L(2, q is the nul immersion defining E in IHr (Le. 'ljJ : E ~ IHr is given by F· F*). The Weierstrass data (g#, w#) of the dual immersion (it is not necessarily an immersion of E, but only of E in general) is given by g# = G, w# = and the metric

--ie,

Bryant Surfaces

95

figure 5

is well defined on E. One proves this is a complete metric on E. Since it is of the form '>"Idzl, where.>.. is the module of a holomorphic (non vanishing) function on E, it follows that E has the right conformal type. 0 Theorem 12.6. Let E be a properly embedded Bryant annular end. Assume E is conformally the punctured disk D* and G extends meromorphicaly to the puncture. Then BooE = G(O). Idea of the proof. Assume G(O) is the point at infinity in the upper half-space model. As in the proof of theorem 12.5, we see that G(O) E BooE. If not E would be contained in some half-space {xi + x~ + x~ < R2, X3 > O} and (as in the proof of theorem 12.5) G would be bounded; a contradiction.

Now suppose E has another point at infinity in its asymptotic boundary, other than G(O) = 00. We can assume this point is 0 = (0,0,0) and qn E E converge to O. Since qn diverge on E, G(qn) -----+ 00, so the mean curvature vector H(qn) at qn is tending to the vertical vector et in direction. In fact for X3(qn) small, E is the graph (w.r.t. the vertical) of a function u and one has the gradient relation:

lV'ul 2 uW 2

2R - u

--w-

Here W 2 = 1 + lV'ul 2 and R is the euclidean radius of the horosphere H(q) of q E E. Now G(O) = 00 so for any b > 0 we can assure that a > b in CT. Then the horizontal component of the normal to the graph satisfies l

=

2

(~) < 2 (~) .

1+(~)2-

b

Hence the normal to the graph can be made arbitrarily close to et = (0,0,1) for u small. If C is the cone with vertex at q, base a disk of fixed radius 3 in

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H. Rosenberg

{X3 = O}, then the graph of u will be above C for b sufficiently big. Thus E is a graph over the base of CT and theorem 12.6 is proved. 0

Now we know that if a Bryant annular end is not dense at infinity, then it is conform ally the punctured disk, G extends meromorphically to the puncture and oooE = G(O). It remains to show that E has finite total curvature (assuming oooE :I Soo). For minimal annular ends in IR3 one shows that finite total curvature implies the right conformal type and the Gauss map g extends to the puncture. For Bryant ends the difficult part is to establish finite total curvature. There are two steps to establish this; the second is the most difficult. First we show that if the end E can be put on one side (the mean convex side) of a catenoid cousin end, then E has finite total curvature. The second step is to find a catenoid cousin end such that E is on its mean convex side. Let us discuss the first case. The mean convex side of a catenoid cousin end is the mean convex component of the end C plus the horospherical (horizontal) disk with the same boundary as C; figure 6.

-

H

c ~~____________________s~=~ figure 6

The end C may be unbounded (i.e. X3 unbounded) or bounded (X3 -+ 0). C is unbounded when C is part of an embedded catenoid and C is bounded when the catenoid is not embedded. When C is unbounded it is easy to show E has finite total curvature. Here is the argument. The end E is determined by F :

E -+

SL(2,Q, F =

(~~),

with

A = Z" j, j holomorphic in D*, and similar representations for B, C, and D. Now X3 = ICI2~IDI2' so if X3 is bounded away from zero on E then ICI and IDI are bounded on E, so they "extend" to the puncture since g = - ~g, the total curvature of E is finite.

Now consider the case when X3 is not bounded away from zero on E. Let z E D* be a conformal parameter for E in which G(z) = }p for some integer p 2: 1, and let r2 = x~ + x~.

Bryant Surfaces

We will assume C (or D) has an essential singularity at contradicts the horosphere theorem.

97

°

and show this

Let ex denote the growth (in r) of the catenoid cousin end below E. Since is an essential singularity of C (in fact, of j, where C = ZV 1), there is some sequence Zn -+ where

°

°

Let qn E E correspond to Zn. Since E is above the catenoid cousin end, for any integer k > 1 and n sufficiently large:

r(qn)

k

> -I Zn-Ip .

That is the horizontal (Euclidean) distance from the point qn to the point s = (0,0,X3(qn)) is at least Iz~lp. Observe that d(qn,G(qn)) is at least d(qn's) - d(G(qn), s) where d denotes the horizontal Euclidean distance. Let I be the horizontal disk of diameter Iz~IP' centered at the point p = (G(qn), X3(qn)). Since the horizontal distance from G(qn) to (0,0) is Iz~lp, the disk I is in the interior of H (qn) +; cf. figure 7.

figure 7

Now the origin is under one of the boundary points of t. Observe that the catenoid cousin C is above the segment [p, s] on I, since the height of C at G(qn) is IG(Ll!" = IZnl pa and X3(qn) ~ IZnl(p+1)a ~ IZnlpa. Since the graph of C is monotone decreasing with r, the segment [p, s] is below C. E is above C so [p, s] is disjoint from E. Moreover let N be a compact embedded surface with boundary the boundary of E so that N U E is an embedded surface. N can be chosen above the catenoid cousin C union the flat disk capping off C. Then exactly as in section 3, N U E separates the ambient space so one can find a path c from s to BE which meets N U E only at the endpoint (first vertical, then a fixed path). Then the k of the above inequality can be chosen large enough so that this path together with the boundary of E is inside H(qn).

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But c together with [p, s] can be extended to a divergent path disjoint from E U H(qn), by going down vertically to G(qn) from p. This divergent path from 8E C H(qn)+ does not meet H(qn) U E again, which contradicts the Horosphere theorem. This completes our discussion of the first case which we state as a theorem. Theorem 12.7. Let E be a properly embedded Bryant annular end. If E is on the mean convex side of a catenoid cousin end, then E has finite total curvature. Now consider finding the catenoid. Theorem 12.8. Let E be a properly embedded Bryant annular end which is not dense at infinity. Then there is a catenoid cousin end C and E is on the mean convex side of C. Proof. We will give a rather detailed discussion of this proof.

We know that 8= E is one point, which we take to be infinity in the upper half-space model of IHf3 . Let B be a ball in IHf3 , whose interior contains 8E and with E transverse to 8B. Let El denote the non compact component of E-B, and let W denote the mean convex domain (along Ed bounded by El and a compact domain on 8B. If X3 ~ C > 0 on E then C can be constructed using a catenoid cousin end below height c which is a graph over an exterior domain xi + x~ > r5, asymptotic to the plane X3 = 0 at infinity. So we can assume there is a sequence qn E El with X3(qn) -7 O. Since 8=E = 00, we have r(qn) = VXl(qn)2 +X2(qn)2 -700. For q E E l , let "( be the minimizing geodesic of IHf3 joining q to a point of 8B. We will be working with q lower than B. Assume B = {xi+X~+(X3-4)2 = I} for convenience, and X3(q) :::; 1, r(q) > 5. Parametrize,,( by arc length so that "((0) is the highest point of "( (which is not on B by our choice of constants), and "((to) = q with to < O. Let pet) be the family of (hyperbolic) planes orthogonal to "( at "((t). For t very negative, pet) is disjoint from El since 8=El = 00, and El is proper so there is a first tl :::; to (as t increases) such that P(tl) touches El at a point ql· We do Alexandrov reflection of El with the planes pet) as t increases from tl to O. Let Set) be symmetry of IHf3 through pet), Edt)+ the part of El on the side of pet) not containing B, and El(t)* = Set) (El(t)+). For t slightly larger than tl, El (t)+ is a graph over (part of) pet), int (El (t)*) C W, and the angle between pet) and El(t)+ is never 7r/2 along 8 E l(t)+. These

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99

properties continue to hold until the first t (t2 say) such that E1 (t2)· touches 8B, for if one of these properties failed to hold at some earlier t, P(t) would be a plane of symmetry of E. Then E is part of a properly embedded mean curvature one compact surface M, with 8M = 0. This is impossible. Clearly t2 plane P(t), t St2@ E B.

< 0 since q is lower than B, so the symmetry of q through some < 0 meets B. Thus there is some point if E E 1(t2)+ such that

Let 15 1 = dist(if, ,), and qt = St({j). Since, is invariant by St, we have dist(qt,,) = 15 1 as well. For t = t2, qt is on :6, so dist(qt,,) ~ diamB = Q. The curve qt joining if to 8B, as t varies from t1 to t2, is an equidistant curve (3 whose distance from, is less than 15, and this equidistant curve is contained in W. We emphasize that this discussion is valid for any q E E1 with X3 (q) < 1, r(q) > 5. In particular, consider the sequence qn E E 1, satisfying X3(qn) -t 0, r(qn) -t Then a subsequence of the geodesics ,n joining qn to B converges to a vertical geodesic over B and the equidistant curves (3n from ifn to B are in Wand a distance at most 15 from ,n' So the equidistant curves (3n are in the tubular neighborhood of,n of radius Q. As n -t 00, the tubular neighborhoods converge to a vertical cone of hyperbolic width Q. Let C(Q) denote this cone; we can assume the base of C(Q) is the origin. 00.

Now we can prove that E1 n A is a graph where A = {X3

< 1, r

~

5}.

Suppose this were not true. Let N be the Euclidean unit normal to E, N . > 0 and suppose that N3 ~ 0 at some point q E E1 n A. Then the horosphere tangent to E at q, H(q), is at most of (Euclidean) radius 1 and

H

8E C H(q)-. Then by theorem 12.4, H(q) separates W into three connected components. One is compact and contains part of 8B. One is non compact, and contains the points ifn, n large. And the third is compact and inside H(q)+. But the equidistant curves (3n are in Wand disjoint from H(q) for n large; this is impossible since the (3n go to 8B in W. This proves E1 n A is a graph. In fact the above argument proves much more: for q E A, H(q) must intersect C (15); otherwise the equidistant curves (3n would be disjoint from H(q) for n large; cf figure 8. For q E E1 n A, let R be the Euclidean radius of H(q) and let d be the Euclidean distance of q to C(Q). Then (since C(Q) is invariant by homothety from (J" and C(Q) n H(q) o:J 0) there is a >. > 0 such that

2R

~

d

~

2>.r(q),

and>' depends only on C(8). In particular R -t

00

when r(q) -t

00.

Now we shall prove that E is below some horosphere X3 = ccmstant. We

100

H. Rosenberg

w

figure 8

know that El n A is the graph of a function u and in theorem 12.6, we derived the formula:

since u

:s 1 this implies 2

l\lul

2u

:s R _ 2 :s

2u

Ar(q) - 2

In particular, at the point qn EEl, where X3(qn) -+ O,r(qn) -+ obtain

for a sequence en -+

00,

we

o.

Now recall our discussion of Alexandrov reflection by planes orthogonal to the geodesics 'Yn joining qn to DB. We found a point qn in E l , associated to the first accident of Alexandrov reflection, and we showed the equidistant curve (3n from q--;' to DB was in W. We have Ir(qn) - r(qn)1 < 1 by construction, so at qn we also have an estimate

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101

Then the maximum oscilation of u on the horizontal (Euclidean) disk D of radius X 3En (Qn), centered at ([n, is 2C:nX3(([n). To check this, notice that the most point of D closest to the origin. So

l'Vul

can be is r(q)-1/2, where q is a

r(q) = r(([n) - X3(([n)r(([n)I/4

~ r(~n).

Then r(q)-1/2 :::; V2r(([n)-1/2 and the oscilation on D is at most

:::; 2X3 (([n)C: n . Define Dn = D + (0,0, X3(([n)); Dn is a horizontal disk above the graph of u over D so Dn C Wand the hyperbolic radius of Dn tends to infinity (it is 1/2c: n ). Also the hyperbolic distance between Dn and the graph of u over D is bounded by In(2). Let tn < 0 denote the first time that S(tn)(([n) touches {JB (the first accident when we do Alexandrov reflection with the planes orthogonal to 'Yn). We have Fn = S(tn)(Dn) C Wand the distance of Fn to {JB is at most In(2). As n -+ 00, Fn converges to a horizontal horosphere F which must be in W. Thus E is below F. Next we observe that E2 = En (D x jR+) is a vertical graph, where D = {xi + x~ > a2 }, for some a > O. To see this, remark that X3(q) :::; Co

Ii

for some constant Co so if (q) does not point up then q is in the upper hemisphere of its tangent horosphere so X3(q) ~ R = the Euclidean radius of H(q). Hence R :::; Co and H(q) will be disjoint from the cone G(8) for r(q) larger than some fixed a. As before, this is impossible since the equidistant curves fin, for n large, willnot intersect H(q). Now on the domain D x jR+ where E is a graph, we consider the family of catenoid cousin ends G(t) with each G(t) a graph over D x jR+ , tangent to the vertical cylinder {JD x jR+ and {JG(t) is at height t on {JD x jR+. These surfaces are described in [10]. For t > Co, {JG(t) is above E. If G(t) intersects E 2, then by theorem 12.9 which we will state below, r = G(t) n E2 is compact. r is not homologous to zero on E2 (nor is any sub cycle of r) since this would yield a compact domain N on E2 whose boundary is in G(t). Now vary t to obtain a last point of contact of G(t) with N; then G(t) = E2 by the maximum principle. It follows that r is a Jordan curve on E2 that generates III (E2). On G(t), r bounds a catenoid cousin end that is below E2 and theorem 12.8 is clear. So we can assume G(t) n E2 = 0 for t > co. Now decrease t to O. There is some largest t where G(t) is disjoint from E2 and G(S)nE2 i= 0, for s < t. Since

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H. Rosenberg

C(t) is vertical along 8n x IR+ and E2 is a graph (not vertical) there, 8C(t) is always above E 2. Thus we are in the previous situation where C(s) n E2 i:- 0 and 8C(s) above E2 and theorem 12.8 is proved.

o Theorem 12.9. Let n be a non compact domain in the plane (Xl, X2) with at least one component of 8n non compact. Let UI, U2 be defined on n with their graphs solutions of the mean curvature equation H = 1 in 1Hf3. Suppose the following conditions are satisfied. a) U2 ::; UI ::; 1 on n, UI = U2 on 8n, CI C2 .. b) ~ ::; U2 ::; ~, for some posztwe constant CI , C 2 , a (U2 is the graph of a r r catenoid cousin), IV'Ul12 C I C C) - - - : : ; 2' Jor some > 0, r 2 -_ Xl2 + x 22· UI

r

It then follows that

UI

= U2

on n.

Remark 12.10. In order to apply this theorem to prove theorem 12.8, we need to verify that the graph u(= UI) of E2 we have in theorem 12.8 satisfies the conditions a,b, and c. The conditions a and b are satisfied by construction; the condition c needs some discussion.

In the proof of theorem 12.6 we derived the gradient bound for

U

(1

u:

IV'uI 2
where R is the Euclidean radius of the horosphere H(q). Since

U ::;

1, we have

IV'uI 2 < _2_. U - R-2 So we need to know R is of order r2 for the graph u, to satisfy condition c. We see this by considering H(q), q on the graph of u. Let E denote the graph of U (this is the E2 in the proof of theorem 12.8), and let C be the vertical compact cylinder joining 8E to the plane X3 = O. Observe that for q E E, H(q) must intersect C. For if H(q) passes over C, then 8E c H(q)- so the figure eight in H(q) n E, would contain a Jordan curve C I that is homologous to 8E on E (theorem 12.4). However, U takes its maximum value on 8E (u has no interior maximum since the graph of U would touch a horizontal horosphere at a local maximum and have the same mean curvature vector). Thus C 1 would be lower than 8E and link the cylinder C. So H(q) must intersect C. We want to estimate 1/ R from above, so for q E E, we can assumme H(q) intersects the vertical segment over the origin at a point p at height X3(P) less than some fixed b > O. Now for X3(q) < b, the horosphere

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103

o

R

b p~=----"----l

G(q)

figure 9

H(q) passing through p, intersects the plane X3 = 0 at the point G(q)j cf. figure 9. Then t 2 + (R - X3(p))2 = R2, so R = 2X~2(p)

+x

3

Jp) ~ ~. Apply the same

;b 2

T

(Pythagorean) calculation with p replaced by q to obtain R ~ where is the horizontal distance from q to G (q). Since T + t ~ r, T or t is at least ~ so

R ~ ~: as desired. We will not elaborate on the proof of theorem 12.9. The techniques used have their origin in the paper of Collin and Krust [8) where they study two graphs u, v over a non compact domain n c ]R2, which satisfy the minimal surface equation, with u = von an. They derive on estimate for the minimum growth of u - v (when u ¥- v), and this technique is used in our proof of theorem 12.9. The reader can also consult the paper by Meeks and Rosenberg where this technique is employed in the form we use in theorem 12.9[26).

13

Non-density at infinity

To complete the proof of our main theorem, it remains to establish that a properly embedded annular Bryant end can not be dense at infinity. First we will discuss how stable Bryant surfaces behave at infinity: they are vertical graphs (with pointing up) and they can not have an open set in their asymptotic boundary, we solve a Plateau problem to obtain stable Bryant surfaces and their position relative to E leads to a contradiction.

11

Stable Bryant surfaces have the property that their Gaussian curvature tends to zero when one is far from the boundary (as the inverse of the square distance to the boundary). This can be seen by considering the isometric minimal cousin in ]R3, which is also stable, and for which one has the desired curvature estimates. Now we will make precise how stable surfaces behave at infinity.

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H. Rosenberg

Let M be a properly embedded mean curvature one surface with aM perhaps not compact. Assume a properly embedded surface ~ exists with o~ = aM and ~ u M = oW with M mean convex along W. Let P be a (hyperbolic) plane with 1Hf3 - P = p+ U P-, the connected components of the complement. Assume ~ C P-. Let M+ = M n p+ . Theorem 13.11. There is a constant c > 0 (independent of M) such that if IK(q)1 < c for q E M+, then int (oooM+) = 0; i.e., M can not be asymptotic to an open set at infinity in P+. In the half-space model, for q E M+ and X3(q) sufficiently small, M+ is a vertical graph near q, no point of M is below this local graph, and the angle between H (q) and et is at most 7r /4. The proof of theorem 13.11 is not difficult and we do not discuss it here. Next, density at infinity is used to obtain an arc on the end having two distinct points at infinity as its asymptotic boundary. It is such an arc which yields a contradiction. More precisely: Theorem 13.12. Let E be a properly embedded Bryant end with oooE = Soo. There is a proper arc 'Y on E with 000 ("((t) It 2: O} = PI, ooo{"((t)lt::; O} = P2 and PI :f:. P2· We give a brief idea how the arc 'Y is obtained. Since E is dense at infinity there are points q on E, lower than oE, where H(q) . et ::; 0 (otherwise E would be a vertical graph near infinity so could not be dense there). Then the horosphere theorem implies (there is some work here) that the connected component of H(q) nE at q is not compact. So one can find a path (in this component) on E from q to G(q) E Soo. Now do the same argument at a point y E E with H(y) n E not compact, to find a path (3 on E joining y to G(y). Choose y so that G(y) :f:. G(q). Then join q to y on E by a path J.l. Then 'Y = 0: U J.l U (3 works. For simplicity, suppose now that the end E is part of a properly embedded Bryant surface M and the arc 'Y of theorem has been constructed on E. Using M as a barrier and solving a Plateau problem for H = 1 surfaces one constructs stable Bryant surfaces [1]. Theorem 13.13. Let M be a properly embedded mean curvature one surface in 1Hf3. Suppose 'Y is a proper arc on M that separates M into two components M I , M 2 • There exists two properly embedded mean curvature one surfaces ~l' ~2 satisfying: are stable, O~l = O~2 = 'Y, ~1 n ~2 = 'Y, bounds a domain R contained in the mean convex component W of 1Hf3 - M, c) R is mean convex, a) b)

~1

and

~1 U ~2

~2

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105

d) L\ U MI separates Iff and E2 U M2 as well. Proof. Here is the idea of this theorem. Let p E 'Y and B = B(R) a geodesic ball of Iff centered at p, of radius R. Let M(R) be the connected component of M n B, containing p and M I , M2 the connected components (containing p) of M(R) - 'Y. MI is homologous, reI 8MI , to a stable Bryant surface EI = EI(R), inside B. One obtains E I , as follows. Consider domains Q in B with 8Q = MI U E, 8E = 8MI , and the functional on (Q,8Q): (Q,8Q) ---+ area(E)

+ 2Vol(Q).

A minimum of this functional exists and EI is smooth Bryant surface in B which is stable. As R -+ 00, a subsequence of the El (R) converge to the stable EI desired. Similarly E2 is constructed with M 2 ; d. figure 10, where D I , D2 are the least area surfaces with boundary 8MI , 8M2 respectively. It is not hard to see that EI U E2 bound a mean convex domain R.

B

figure 10

o We will now show stable surfaces like this can not exist.

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H. Rosenberg

Let r c Soo = {X3 = o} U { 00 }, be a circle separating PI and P2' 171 and 172 are stable so their curvature is small far from ,. In particular, when C is the constant of theorem 13.11, there is a Co > 0 such that IK(q)1 < c for q E 17 = 171 U 172, dist(q,,) 2: Co. Let T be those points of IHf3 whose Euclidean distance to r is at most > O. Then for C1 sufficiently small, 171 n T and 172 n T are vertical graphs over domains ill and il2 C {X3 = O}. We know H~ > 0 on 9 = (171 n T) U (17 2 n T) and 171 U 172 bounds a mean convex domain R by theorem 13.13, so ill n il2 = 0. Also we can assume the angle between H(q) and ~ is less C1

than 7f / 4 on g. Then for q E g, and t = X3 (q) sufficiently small, g, near q, is a vertical graph over a horizontal disk D(q), centered at q, of Euclidean radius

t. Let t > 0 and rt = r + t~. Choose t small so that rt c T and rt transverse to 17. The linking number of rt and, is one so rt n 171 consists an odd number of points. 17 = 171 U 172 bounds the mean convex domain so there is an arc of rt , which we denote (q1,q2), joining a point q1 E 171 q2 E 172 and the interior of the arc is in the interior of R.

is of R to

For q on the arc (q1,q2), let J(q) be the disk D(q) together with the lower hemisphere of the horosphere that contains 0 D (q) and is vertical along 0 D (q). J(q) has a corner along oD(q). For q = q2, J(q) C il2 X IR+ by our gradient bound on the graph g. Now move q on the arc (q1, q2) from q2 to q1. We know that ill n il2 = 0 so J(q2) n 171 = 0. There will be a first q on the arc where J(q) touches 17 1. We will next see that J(q) touches 171 at infinity. Suppose J(q) first touches 171 at a smooth point P on the horosphere in J(q). The mean curvature vector of the horosphere points up at p, and the mean curvature vector of 171 points up at P too. So the vectors are equal and 171 is a horosphere. This is impossible because the proper arc, is on 171 and r has two points at infinity PI and P2; the horosphere has one point at infinity. Next suppose the first point P where J(q) touches 171 is on oD(q). We know that the horizontal segment in D(q), joining P to q (which we call [p, q]) meets 171 only at p. Also this segment does not meet 172 because our gradient bound implies 171 is a graph over D(p); J(p) C ill X IR+. Thus the segment [p, q] is contained in R. to 171 at p is a support plane of J(q) and H(p) the fact that R is mean convex: 11 (p) points on the other side of the tangent plane than

The (Euclidean) tangent plane points up at p. This contradicts into R, and [p, q] C R, [p, q] is (p).

H

Thus there is a point g on the arc where J(q) touches 171 for the first time at a point goo E r; cf. figure II. Now consider q' on the arc (q, q2) at Euclidean distance less than t/2 from q, such that the point of r below q' is not in 0 00 171 but qoo is below D(q'). By

Bryant Surfaces

107

q I

J(q)

r

figure 11

lemma 13.14, which follows, there exists a one parameter family of vertical graphs C (t), 0 < t ~ 1, such that C (1) is the original horosphere of J( q'), and C(t) (t < 1) is a catenoid cousin end; each C(t) is vertical along aC(t) and aC(t) is contained in the vertical cylinder aD(q') x IR+. As t -+ 0, x3ic(t) -+ o. Since 171 is a graph in this cylinder, C(t) can not meet 171 for the first time at a point of aC(t) (where C(t) is vertical). C(t) can not touch 171 at an interior point by the maximum principle, nor at infinity. So C(t) never touches 171 and qoo can not be in the asymptotic boundary of 171 . This argument together with theorems 13.11, 13.12, 13.13 and lemma 13.14, proves that a properly embedded Bryant annular end cannot be dense at infinity. Lemma 13.14. Let C be a circle in {X3 = O} with center qoo = (0,0). There is a one parameter family of catenoid cousin (and horosphere) ends C(t), o < t ~ 1, satisfying: a) each C (t) is a vertical graph over {O b) C(t) is vertical over {x2 +y2 = a 2 }, c) x3(C(1)) = a, C(l) is a horosphere, d) qoo = aooC(t), for each t, and e) X3(C(t)) -+ 0 as t -+ 0,

f) H(C(t)).~ 2:

< X2 + y2 < a 2 },

a the radius of C,

o.

Proof. Gomes has proved that a family of this nature exists as graphs over the exterior domain of C [10]. To get the C(t) of the lemma, one does inversion of this family through a plane P with aooP = C, followed by a homothety from qoo; cf. figure 12; the homothety takes b to a.

o Theorem 13.15. Let E be a properly embedded annular end of mean curvature one. Then E is not dense at infinity, has finite total curvature and is regular. Hence E is asymptotic to a catenoid cousin end or to a horosphere end.

108

H. Rosenberg I rotation

p

qoo

C(t)

figure 12

Proof. We have already proved the first part of theorem 13.15 when E is a part of a properly embedded Bryant surface M. Here is the argument in general. Let E be a compact embedded surface such that oE = oE and M = EUE is an embedded surface (not necessarily smooth along oE). Change the metric of JHf3 in a compact neighborhood of E so that M has mean curvature greater than 1 near E. Now do the previous arguments with M in this new metric. The E I , E2 one obtains will satisfy all the conditions necessary to do the argument as before. What matters is the structure of E I , E2 near infinity. The same arguments then show E can not be dense at infinity. When E is a finite total curvature Bryant annular end, which is regular and embedded then R. Earp and E. Toubiana have shown that is asymptotic to a horosphere or a catenoid cousin end [35]

o 14

Some applications of the annular end theorem

Theorem 14.16. Let M be a properly embedded Bryant surface in JHf3 of finite topology. If M is simply connected, M is a horosphere. If Mis 1-connected then M is a catenoid cousin. If M has three ends, then M is a bigraph over a plane P, i.e., M is invariant by symmetry in P and each component of M - P is a geodesic graph over P.

Proof. When M is simply connected, oooM is one point by theorem 13.15. Then do Carmo and Lawson [9] proved M is a horosphere. When M is 1connected, oooM is two points and M is invariant by rotations about the geodesic joining the two points [24]. Thus M is a catenoid cousin. When M

Bryant Surfaces

109

has three ends, 800 M consists of three points so 800 M is contained in a circle of Soo. The conclusion is then proved in [24].

o Theorem 14.17. Let M be a properly embedded mean curvature one surface in JH[3, M not a horosphere. Then each annular end of M is asymptotic to a catenoid cousin end. Proof. We know by theorem 13.15, that each annular end E is asymptotic to a catenoid end or to a horosphere end. We will assume E is asymptotic to a horosphere end and obtain a contradiction. We work in the upper half-space model of JH[3, {X3 > O}, and assume E is asymptotic to a horosphere X3 = C > o. In particular the mean curvature vector of E points up outside of some compact set of E. There are no ends of M above E since their mean curvature vector would also point up ( each such end is asymptotic to a horizontal horosphere or a catenoid cousin end whose limiting normal points vertically up) and M separates JH[3 into two connected components so no such end is above E. Then for c > 0, the part A of M above c + c is compact. At the highest point of A (if A were not empty) the mean curvature vector of M points down. But this highest point can be joined by an arc in JH[3 - M, to a point of E where the mean curvature vector points up. Thus M is completely below X3 = c. Let c > 0 and let 0 be a small circle in the plane X3 = c - c so that is above M. Just as in the proof of the half-space theorem for properly immersed minimal surfaces in JH[3 [29], one can take a family of catenoid cousin ends 0('\), 80(1) = 0 with 0(1) above M, and 0('\) converges to the plane X3 = c - cas ,\ -+ O. Then some 0('\) would touch M at a point q E M and the maximun principle would yield M = this catenoid cousin. Thus each end of M is asymptotic to a catenoid cousin. 0

o

References 1. H.Alencar and H.Rosenberg, Some remarks on the existence of hypersurfaces of constant mean curvature with a given boundary, or asymptotic boundary, in hyperbolic space; Bull. Sci. math. 121 (1997),p61-69. 2. R.Bryant, Surfaces of mean curvature one in hyperbolic space; Asterisque 154-155 (1987), Soc. Math de France, p.321-347. 3. L. Barbosa and R.SaEarp, preprint. 4. P. Collin, Topologie et courbure des surfaces minim ales proprement plongees de ]R3; Ann. of Math. 2nd Series. 145 (1997), p.1-31. 5. A.!, Bobenko, All constant mean curvature tori in ]R3, §3, JH[3 in terms of theta functions; Annallen, (1991) 209-245. 6. H.Choi and R.Schoen, The space of minimal embeddings of a surface into a three-dimensional manifold of positive Ricci curvature; Invent. math. 81, p.387-394, (1985).

110

H. Rosenberg 7. P.Collin, L.Hauswirth and H.Rosenberg, The geometry of finite topology surfaces properly embedded in hyperbolic space with constant mean curvature one; to appears in th Ann. of Mat .. 8. P. Collin, R Krust: Le probleme de Dirichlet pour I'equation des surfaces minim ales sur des domaines non barnes; Bull. Soc. Math. France 119 (1991),443-462. 9. M. do Carmo and B.Lawson, On Alexander-Bernstein theorems in hyperbolic space; Duke Math. J. 50 (1983), p.995-1003. 10. J.M. Gomes, Spherical surfaces with constant mean curvature in hyperbolic space; Bol. Soc. Bras. Mat. 18 (1987), p.49-73. 11. RHardt and H.Rosenberg, Open book structures and unicity of minimal submanifolds Ann. l'instit. Fourier, 40 (1990), 701-708. 12. D.Hoffman, How to use a computer to find new minimal surfaces;Journee Annuelle, Soc. Math. France, (juin 93),1-29. 13. D.Hoffman and W.H.Meeks III, The strong half-space theorem for minimal surfaces, Invent. Math. 101(1990), 373-377. 14. H.Hopf, Differential geometry in the large; Lecture Notes. Math. Springer Verlag, 1000 (1983). 15. N.Kapouleas, Complete constant mean curvature surfaces in Euclidean three-space; Annals of Math. 131 (1990), 239-330. 16. N.Kapouleas, Compact constant mean curvature surfaces in Euclidean three space; J. Diff. Geom. (1991),683-715. 17. N.Kapouleas, Complete embedded minimal surfaces of finite total curvature, preprint. 18. H.Karcher, The triply periodic minimal surfaces of Alan Schoen and their constant mean curvature companions; Manuscripta Math. 64, (1989), 291-357. 19. H. Karcher, Hyperbolic constant mean curvature one surfaces with compact fundamental domains., preprint. 20. N.Korevaar, RKusner, W.H. Meeks III and B.Solomon,Constant mean curvature surfaces in hyperbolic space, Amer. J. Math. 114 (1992), 1-43. 21. N.Korevaar, R.Kusner and B.Solomon, The structure of complete embedded surfaces with constant mean curvature; J. Diff. Geom. 30 (1989), 465-503. 22. B.Lawson, Complete minimal surfaces in §3 Annals of Math. 92,(1970), 335-374. 23. B.Lawson, The unknottedness of minimal embeddings Invent. Math. 11 (1970),183-187. 24. G.Levitt and H.Rosenberg, Symmetry of constant mean curvature hypersurfaces in hyperbolic space; Duke Math. J. 52 (1985), p.53-59. 25. W.H.Meeks III, The topology and geometry of embedded surfaces of constant mean curvature; J.Diff. Geom. 27 (1988), 539-552. 26. W.H. Meeks III and H.Rosenberg, The geometry and conformal structure of properly embedded minimal surfaces of finite topology in JR3; Invent. Math. 114 (1993), 625-639. 27. B.Nelli and H.Rosenberg, Some remarks on positive scalar and GaussKronecker hypersurfaces of lRn + 1 and JH["+1 , Annals of Institut Fourier 47(1997), 1209-1218. 28. U.Pinkhall-I.Stirling, On the classification of constant mean curvature tori; Annals of Math. 130 (1989), 407-451.

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29. L.Rodriguez and H.Rosenberg, Half-space theorems for mean curvature one surfaces in hyperbolic space, Proc. A.M.S 126 1998, 2763-2771. 30. H.Rosenberg, Some recent developments in the theory of properly embedded minimal surfaces in R3 , Asterique, 206 (1992), 463-535. 31. M.Traizet, Construction de surfaces minim ales en recollant des surfaces de Scherk, Annals of Institut Fourier, 46 1996, 1385-1442. 32. M.Umehara and K.Yamada, Complete surfaces of constant mean curvature 1 in the hyperbolic 3-space; Annals of Math 137 (1993). 33. M.Umehara and K.Yamada, A duality on CMC-l surfaces in hyperbolic space and a hyperbolic analogue of the Osserman inequality; Tsukaba J. Math 21(1997), 229-237. 34. W.Rossman, M.Umehara and K.Yamada, Irreducible constant mean curvature 1 surfaces in hyperbolic space with positive genus, T6hoku Math J. 49 (1997), 449-484. 35. R.Sa Earp and E.Toubiana, Remarks on the geometry of constant mean curvature one surfaces in hyperbolic space; preprint. 36. H.C.Wente, Counterexample to a conjecture of H.HopJ, Pacific J. Math. 121(1986), 193-242. 37. Zu-Huan Yu, The value distribution of the hyperbolic Gauss map; Proceeding of the american mathematical society 125(1997), 2997-3001.

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114

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115

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1981

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116

1994

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120. Recent Mathematical Methods in Nonlinear Wave Propagation 121. Dynamical Systems 122. Transcendental Methods in Algebraic Geometry

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1995

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Vol. 1661: A. Tralle, J. Oprea, Symplectic Manifolds with no Kahler Structure. VIII, 207 pages. 1997. Vol. 1662: J. w. Rutter, Spaces of Homotopy SelfEquivalences - A Survey. IX, 170 pages. 1997. Vol. 1663: Y. E. Karpeshina; Perturbation Theory for the Schrodinger Operator with a Periodic Potential. VII, 352 pages. 1997. Vol. 1664: M. Vath, Ideal Spaces. V, 146 pages. 1997. Vol. 1665: E. Gine, G. R. Grimmett, L. Saloff-Coste, Lectures on Probability Theory and Statistics 1996. Editor: P. Bernard. X, 424 pages, 1997. Vol. 1666: M. van der Put, M. F. Singer, Galois Theory of Difference Equations. VII, 179 pages. 1997. Vol. 1667: J. M. F. Castillo, M. Gonzalez, Three-space Problems in Banach Space Theory. XII, 267 pages. 1997. Vol. 1668: D. B. Dix, Large-Time Behavior of Solutions of Linear Dispersive Equations. XIV, 203 pages. 1997. Vol. 1669: U. Kaiser, Link Theory in Manifolds. XIV, 167 pages. 1997. Vol. 1670: J. W. Neuberger, Sobolev Gradients and Differential Equations. VIII, 150 pages. 1997. Vol. 1671: S. Bouc, Green Functors and G-sets. VII, 342 pages. 1997. Vol. 1672: S. MandaI, Projective Modules and Complete Intersections. VIII, 114 pages. 1997. Vol. 1673: F. D. Grosshans, Algebraic Homogeneous Spaces and Invariant Theory. VI, 148 pages. 1997. Vol. 1674: G. Kiaas, C. R. Leedham-Green, W. Plesken, Linear Pro-p-Groups of Finite Width. VIII, 115 pages. 1997. Vol. 1675: J. E. Yukich, Probability Theory of Classical Euclidean Optimization Problems. X, 152 pages. 1998. Vol. 1676: P. Cembranos, J. Mendoza, Banach Spaces of Vector-Valued Functions. VIII, 118 pages. 1997. Vol. 1677: N. Proskurin, Cubic Metaplectic Forms and Theta Functions. VIII, 196 pages. 1998. Vol. 1678: O. Krupkova, The Geometry of Ordinary Variational Equations. X, 251 pages. 1997. Vol. 1679: K.-G. Grosse-Erdmann, The Blocking Technique. Weighted Mean Operators and Hardy's Inequality. IX, 114 pages. 1998. Vol. 1680: K.-Z. Li, F. Oort, Moduli of Supersingular Abelian Varieties. V, 116 pages. 1998. Vol. 1681: G. J. Wirsching, The Dynamical System Generated by the 3n+l Function. VII, 158 pages. 1998. Vol. 1682: H.-D. Alber, Materials with Memory. X, 166 pages. 1998. Vol. 1683: A. Pomp, The Boundary-Domain Integral Method for Elliptic Systems. XVI, 163 pages. 1998. Vol. 1684: C. A. Berenstein, P. F. Ebenfelt, S. G. Gindikin, S. Helgason, A. E. Tumanov, Integral Geometry, Radon Transforms and Complex Analysis. Firenze, 1996. Editors: E. Casadio Tarabusi, M. A. Picardello, G. Zampieri. VII, 160 pages. 1998. Vol. 1685: S. Konig, A. Zimmermann, Derived Equivalences for Group Rings. X, 146 pages. 1998. Vol. 1686: J. Azema, M. Emery, M. Ledoux, M. Yor (Eds.), Seminaire de Probabilites XXXII. VI, 440 pages. 1998. Vol. 1687: F. Bornemann, Homogenization in Time of Singularly Perturbed Mechanical Systems. XII, 156 pages. 1998. Vol. 1688: S. Assing, W. Schmidt, Continuous Strong Markov Processes in Dimension One. XlI, 137 page. 1998.

Vol. 1689: W. Fulton, P. Pragacz, Schubert Varieties and Degeneracy Loci. XI, 148 pages. 1998. Vol. 1690: M. T. Barlow, D. Nualart, Lectures on Probability Theory and Statistics. Editor: P. Bernard. VIII, 237 pages. 1998. Vol. 1691: R. Bezrukavnikov, M. Finkelberg, V. Schechtman, Factorizable Sheaves and Quantum Groups. X, 282 pages. 1998. Vol. 1692: T. M. W. Eyre, Quantum Stochastic Calculus and Representations of Lie Superalgebras. IX, 138 pages. 1998. Vol. 1694: A. Braides, Approximation of FreeDiscontinuity Problems. XI, 149 pages. 1998. Vol. 1695: D. J. Hartfiel, Markov Set-Chains. VIII, 131 pages. 1998. Vol. 1696: E. Bouscaren (Ed.): Model Theory and Algebraic Geometry. Xv, 211 pages. 1998. Vol. 1697: B. Cockburn, C. Johnson, C.-W. Shu, E. Tadmor, Advanced Numerical Approximation of Nonlinear Hyperbolic Equations. Cetraro, Italy, 1997. Editor: A. Quarteroni. VII, 390 pages. 1998. Vol. 1698: M. Bhattacharjee, D. Macpherson, R. G. Moller, P. Neumann, Notes on Infinite Permutation Groups. XI, 202 pages. 1998. Vol. 1699: A. Inoue, Tomita-Takesaki Theory in Algebras of Unbounded Operators. VIII, 241 pages. 1998. Vol. 1700: W. A. Woyczy 'ski, Burgers-KPZ Turbulence, XI, 318 pages. 1998. Vol. 1701: Ti- Jun Xiao, J. Liang, The Cauchy Problem of Higher Order Abstract Differential Equations, XII, 302 pages. 1998. Vol. 1702: J. Ma, J. Yong, Forward-Backward Stochastic Differential Equations and Their Applications. XIII, 270 pages. 1999. Vol. 1703: R. M. Dudley, R. NorvaiSa, Differentiability of Six Operators on Nonsmooth Functions and p-Variation. VIII, 272 pages. 1999. Vol. 1704: H. Tamanoi, Elliptic Genera and Vertex Operator Super-Algebras. VI, 390 pages. 1999. Vol. 1705: I. Nikolaev, E. Zhuzhoma, Flows in 2-dimensional Manifolds. XIX, 294 pages. '999. Vol. 1706: S. Yu. Pilyugin, Shadowing in Dynamical Systems. XVII, 271 pages. 1999. Vol. 1707: R. Pytlak, Numerical Methods for Optimal Control Problems with State Constraints. XV, 215 pages. 1999. Vol. 1708: K. Zuo, Representations of Fundamental Groups of Algebraic Varieties. VII, 139 pages. '999. Vol. 1709: J. Azema, M. Emery, M. Ledoux, M. Yor (Eds), Seminaire de Probabilites XXXlII. VIII, 418 pages. 1999. Vol. 1710: M. Koecher, The Minnesota Notes on Jordan Algebras and Their Applications. IX, 173 pages. 1999. Vol. 1711: W. Ricker, Operator Algebras Generated by Commuting Projections: A Vector Measure Approach. XVII, 159 pages. 1999. Vol. 1712: N. Schwartz, J. J. Madden, Semi-algebraic Function Rings and Reflectors of Partially Ordered Rings. XI, 279 pages. 1999. Vol. 1713: F. Bethuel, G. Huisken, S. Miiller, K. Steffen, Calculus of Variations and Geometric Evolution Problems. Cetraro, 1996. Editors: S. Hildebrandt, M. Struwe. VII, 293 pages. 1999. Vol. 1714: O. Diekmann, R. Durrett, K. P. Hadeler, P. K. Maini, H. L. Smith, Mathematics Inspired by Biology. Mar-

tina Franca, 1997. Editors: V. Capasso, O. Diekmann. VII, 268 pages. 1999. Vol. 1715: N. V. Krylov, M. Rockner, J. Zabczyk, Stochastic PDE's and Kolmogorov Equations in Infinite Dimensions. Cetraro, 1998. Editor: G. Da Prato. VIII, 239 pages. 1999. Vol. 1716: J. Coates, R. Greenberg, K. A. Ribet, K. Rubin, Arithmetic Theory of Elliptic Curves. Cetraro, 1997. Editor: C. Viola. VlII, 260 pages. 1999. Vol. 1717: J. Bertoin, F. Martinelli, Y. Peres, Lectures on Probability Theory and Statistics. Saint-Flour, 1997. Editor: P. Bernard. IX, 291 pages. 1999. Vol. 1718: A. Eberle, Uniqueness and Non-Uniqueness of Semigroups Generated by Singular Diffusion Operators. VlII, 262 pages. 1999. Vol. 1719: K. R. Meyer, Periodic Solutions of the N-Body Problem. IX, 144 pages. 1999. Vol. 1720: D. Elworthy, Y. Le Jan, X-M. Li, On the Geometry of Diffusion Operators and Stochastic Flows. IV, 118 pages. 1999. Vol. 1721: A. Iarrobino, V. Kanev, Power Sums, Gorenstein Algebras, and Determinantal Loci. XXVII, 345 pages. 1999. Vol. 1722: R. McCutcheon, Elemental Methods in Ergodic Ramsey Theory. VI, 160 pages. '999. Vol. 1723: J. P. Croisille, C. Lebeau, Diffraction by an Immersed Elastic Wedge. VI, 134 pages. '999. Vol. 1724: V. N. Kolokoltsov, Semiclassical Analysis for Diffusions and Stochastic Processes. VlII, 347 pages. 2000. Vol. 1725: D. A. Wolf-Gladrow, Lattice-Gas Cellular Automata and Lattice Boltzmann Models. IX, 308 pages. 2000.

Vol. 1726: V. Marie, Regular Variation and Differential Equations. X, 127 pages. 2000. Vol. 1727: P. Kravanja M. Van Barel, Computing the Zeros of Analytic Functions. VII, III pages. 2000. Vol. 1728: K. Gatermann Computer Algebra Methods for Equivariant Dynamical Systems. Xv, 153 pages. 2000. Vol. 1729: J. Azoma, M. Emery, M. Ledoux, M. Yor Seminaire de Probabilites XXXIV. VI, 431 pages. 2000. Vol. 1730: S. Graf, H. Luschgy, Foundations of Quantization for Probability Distributions. X, 230 pages. 2000. Vol. 1731: T. Hsu, Quilts: Central Extensions, Braid Actions, and Finite Groups. XII, 185 pages. 2000. Vol. 1732: K. Keller, Invariant Factors, Julia Equivalences and the (Abstract) Mandelbrot Set. X, 206 pages. 2000. Vol. 1733: K. Ritter, Average-Case Analysis of Numerical Problems. IX, 254 pages. 2000. Vol. 1734: M. Espedal, A. Fasano, A. Mikelie, Filtration in Porous Media and Industrial Applications. Cetraro 1998. Editor: A. Fasano. 2000. Vol. 1735: D. Yafaev, Scattering Theory: Some Old and New Problems. XVI, 169 pages. 2000. Vol. 1736: B. O. Turesson, Nonlinear Potential Theory and Weighted Sobolev Spaces. XIV, 173 pages. 2000. Vol. 1737: S. Wakabayashi, Classical Microlocal Analysis in the Space of Hyperfunctions. VIII, 367 pages. 2000. Vol. 1738: M. Emery, A. Nemirovski, D. Voiculescu, lectures on Probability Theory and Statistics. XI, 356 pages. 2000.

Vol. 1739: R. Burkard, P. Deutlhard, A. Jameson, J.-L. Lions, G. Strang, Computational Mathematics Driven by Industrial Problems. Martina Franca, 1999. Editors: V. Capasso, H. Engl, J. Periaux. VII, 418 pages. 2000.

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