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• M (c) with flat normal bundle if and only if dQx + Qx A 0 A = 0 for A € C* and 61, • • •, 6n are linearly
(9)
independent.
When the isometric immersion tp has flat normal bundle, the second fundamental of (p can be simultaneously diagonalized. In such a case, we can choose the tangent frame fields je*} to Mn(c) such that Wj, n+Q = bia6l. Moreover, we can choose a parallel normal frame fields {eQ} so that rj = 0. On putting k
we have from (2), (4) and (7) ^2 kabja + EV2 = 0,
(i ^ j)
a
(bia - bja)T^ = {bia - bka)Tjik, ej{bia) = (bja - bicJFij.
(i,j,k
(i ^ j)
#
(10)
95 Set y = (tv-^)r,
fi = (6te),
n
Js=^~n
B = (B V), Then we see from (8) that BJgBT
°).
(11)
= diag(/9i, • • • pn), where
Pi = £ ^
+ £ V
"
(12)
L e m m a 2.2 Let <^ : Mn(c) D U -> M (c) oe a locally isometric immersion with flat normal bundle. Assume that Pi ^ 0 for all i where pi are smooth functions defined by (12). Then there exist a line of curvature coordinates (xi) on U such that the first and second fundamental forms of (f can be given by
i
II = ^2oJfbiadxfen+a.
(13)
Proof. Since Pi ^ 0 then we can write pi = ±(a;) follows from (10) and (12) ejiai) = aiTlj,
2
with a; > 0. It
(t#j).
(14)
For any point x e U, if we choose e n + i at x so that bn(x) ^ 0, 612(2;) = • • • = bi,m-n(x) = 0, then it follows from (10)i and (12) that M z ) = - 7 - T T (*? £ 1 )' &ii(*) + e>2 = ^ 0 . (15) On (a;) By taking a — j — 1 in (10)2, we see from (15) that T^(x) = 0. Since x is arbitrary, then it follows that T*j = 0 for i, k, 1 distinct. We know that the components T^ of Wjj are independent of the choice of the fields of normal frames. Thus, we have T*- = 0 for i,j, k distinct. Hence, by using (14) and the skew-symmetry, we conclude that Wi. =
e
e
jM6i _ Oj
-iMej.
(16)
Oj
As the same as in [4], it is easy from (16) to see that there exist a line of curvature coordinates (x») on U so that d/dxi = a;ej, 9l = aidxi. Thus, the first and second fundamental forms of ip are given by (13).
96
In the case that e > 0, we see from (10) that rankZ? = n, which implies that m — n > n — 1 and Pi > 0 for all i, so that the conclusion of Lemma 2.2 holds. In the case that e = 0, if m — n > n and the normal bundle of
M (c) be a non-umbilically local isometric immersion with flat normal bundle. Suppose that $\ is a solution of the system (28), /x € R \ {0}, and L is a real constant s x (2n + 1) matrix satisfying (39). Let h — LTQ^/^J^ and $A = $\D\ where D\ is the Darboux matrix given by (36). If o,-(0) ^ 0, for all j , then there exist an open neighborhood U around the origin O € M™(c) and a non-umbilically 2n—1
local isometric immersion (p : U -> M (c) such that f = re ° (p can be expressed explicitly by f = QD^iO)*™ = QD^WQtD™, (44)
103 where Q is a constant matrix denned by (31) with A(0). Remark 1. The above process of the Darboux transformation is purely algebraic. Hence, starting from a special solution A to (28) for which the corresponding r = Q$[ ' may be degenerated, we can repeat the processes via the purely algebraic algorithm and obtain a sequence of solutions to (28): $ A -> $ A - • $ A -> • • •, from which we obtain a sequence of local 2n—1 isometric immersions from Mn(c) to M (c). Remark 2 The method here may be used to study local isometric immersions from Riemannian products of space forms into space forms. We are going to discuss these problems in a forthcoming paper.
5
The construction of local isometric immersions derived from a trivial solution
By take a trivial solution of (25) as F = 0,
A = In,
6 =
(0,---,0,KO,
We solve (28) to get /
fc];'1
0
0
Xi
0 "X*
$>
n
0
\J
•••0
• •• u
Yn-i Xn
-Yt
-y„-i
0
Xn-i sxYn
0
0 Ay Xr x 0
Xi
•••
e^$A! ' m \
C
0
Yi
X„-i
o ,ro £9 e*l X
-f£Y "X n 0
n
0
\
•••0
0 $ in,tit
where
•i' 1 = ^ ( « a * « + e _ A 2 ) '
*rm =
^xn+^),
,
(45)
104 K\
"A.
Xa = cos(Aa;a),
Xn = cos(xz n ),
Ya = sin(Ax a ),
Yn = sin(xx„).
Choose fj, € R and i = (lo, h, • • •, hn) 6 R2n+1 5 Z '"+« + a 2 n = £ ' ? + ei o ^ 0, a
such that
«Z2n - A*i//0 = 0.
(46)
j
It is easily seen that h =
/T$V^TM
= (f,
JJ,
V^IC) = (€.i7i»---.»?n,V/::TCi,---,"v/=:TCn)
satisfies (38) and (40). From (41), (42) and (43) we know that
"i"" ( O T (c + ^ - ^ , - ^ „ f «*)*, A = £
C* + eCn,
A = J n - I JeCTC-
(47)
a
By Theorem 4.4, We can use directly the formula f = Q£>j"1(0)$-Di in a neighborhood U of the origin. Such f is nondegenerate only if there exists a point x £ U such that bj(x) ^ 0 for all j . For this aim, we need only to choose suitably I such that ln+j^0,
£ £
+
a + (£-2)JL^0.
(48)
a
It may guarantee that bj(0) ^ 0 for all j . Then there exists an open neighborhood U of O such that f is nondegenerate in U, which implies that -
2n—1
there is a local isometric immersion ip : U -> M (c) so that f = r g o (p. Moreover, by using $Ai we can obtain a new solution $ A of (28). Continuing this process, a series of local isometric immersions are obtained via a purely algebraic algorithm. For simplicity, in the case that 7 = y/en2 — EKV~X 6 R, we may take I = (lo,h,m •-,'n-i>7' / K ~ '0>'1 j • • • Jn), where ln = IIVK^IQ, IJ ^ 0 for all j , and £ ) a ^ + (^ _ 2)/£ ^ 0. It is clear that (46) and (48) are satisfied. Then we have from (45), (47) and (43) £ = l0e~^
= -Cn,
105 'la — l a c
— sco
6Q = - | W n e ^ ° + 7 1 " ,
(49)
a
In the case that 7 € ^/^Ti?, we may take Z = ( Z o , i l , - - - , J n - 2 ) 0 , 0 , Z i , - - - , Z n _ 2 , V—l7/i"~
In,In),
where Zn = HVK~1IQ, Z Q _I ^ 0 for all a, and ^ I " ^ 2 l? + (e-2fj?v2K~2)ll ^ 0. The remainder is similar to the above. Example n — 2. c = e = /i = v = K = l. c — 0, E = —1, x = ^/T^A2, 7 = v^=2. By taking I = (Zo,0,0,-\/2Zo,Zo) with Z0 ^ 0, we have from (45), (47) and (43) £ = C2 = Zo cos V2x2, t]2 =-losinV2x2,
m = -V2lo sinh xi, Ci = - v ^ / o c o s h x i ,
/
f=-^f
v 2 cos v2a;2 (sinh Xi cos xi + cosh Xi sin xi) a;2A+^sin2v^x2 \ \/2A + v^cosv^2x2(shxi sinxi - chxi cos^i)
7 _ 1 ( -2ch 2 xi - cos2 X2 ~ A V -2v / 2cha;i cos y/2x2
2-\/2cha;i cos y/2x2 2ch 2 xi + cos2 y/2x2
where A = 2ch2a;i - cos2 v2x2.
Thus, f is a piece of non-umbilical surface with constant Gauss curvature 1 in R3. Acknowledgments Project supported by the National Natural Science Foundation of China, the Scientific Foundation of the National Education Department of China, and the Natural Science Foundation of Zhejiang Province.
106
References [1] D. Ferus and F. Pedit, Math. Ann. 305, 329(1996). [2] D. Hilbert, Trans. AMS2, 87(1901). [3] Q. He and Y. B. Shen, Explicit construction for local isometric immersions of space forms, (preprint). [4] J. D. Moore, Pacific J. Math. 40, 157(1972). [5] F. Pedit, Comm. Math. Helv. 63, 672(1988). [6] C. L. Terng, J. Diff. Geom. 45, 407(1997). [7] C. L. Terng and K. Uhlenbeck, Comm. Pure and Appl. Math.53, 1(2000). [8] F. Xavier, Comm. Math. Helv. 60, 280(1985). [9] Z. X. Zhou, Inverse Problems 14 , 1353(1998).
On the Nirenberg Problem Ji, Min (institute of mathematics,
Academia Sinica, Beijing 100080, China)
[email protected]
Abstract. In this paper we strengthen the Kazdan-Warner necessary condition so as to be necessary and sufficient for the Nirenberg problem. Enlightened by this condition, we establish a degree theory which improves some important results previous and gives some results completely new.
§1.
Introduction
We begin with the prescribed scalar curvature problem on a general Riemannian manifold (M, go) which is compact and of dimension two. Given a continuous function R on M, whether can R be the scalar curvature of some metric g which is pointwise conformal to the original metric
- Agou
+ Ro-Reu
= 0,
on
M,
where i?o is the scalar curvature of the metric go • It is related to topology of the manifold M. Integrating the equation (1.1) and using Gauss-Bonnet theorem, we see [ ReudA = 4irx(M) JM where dA is the volume element of (M, g0) and x(M) is the Euler-Poincare characteristic. Clearly (1.2) requires different conditions of R for different signs of x(M). (1.2)
* Supported in part by the NNSF of China(Grant No. 19725102) and the 973 project of China.
107
108
In the case x(M) < 0, ones have had a good knowledge for the solvability. For example, it is known that (1.1) always admits a solution if R < 0 (cf. [19]). In the case x(-^0 = 0, the problem was solved by M.S. Berger and Kazdan-Warner, who gave the following necessary and sufficient condition(Theorem 6.1 in [19]) B - K - W Theorem. Suppose x(-W) = 0. The equation (1.1) is solvable if and only if either (1) R = 0, or (2) R changes sign and satisfies E 0 n BKW jt 0, where So =
se
t of the solutions to A 9o u — Ro,
and BKW = {u&H1(M):
f
R eudA < 0}.
JM
In the case x(-W) > 0, there are only two possibilities that M = S2 with x(-W) = 2 and M — RP2 , the real projective space, with x(M) = 1. From (1.2) we see easily a necessary condition that R is positive somewhere, which is proved also sufficient for the latter case(cf. [1], [22]). Thus, the former case M = S2 is only remaining and also difficult. If (M, g0) is the standard 2-sphere (S 2 , go), the problem is called the Nirenberg problem. Now Let S2 = {x = (xi,x2,x3)
e R3 : x\ + x\ + x\ = 1}
with the natural metric go — dx\ + dx\ + dx\. Let y a n d A be the gradient and the Laplacian operator on (S2,go) respectively. Actually the Nirenberg problem is to determine which smooth function R > 0 can be the scalar curvature of a conformal metric g — eugo (see [20]), i.e. to determine function R > 0 such that the following PDE: (1.3)
-Au
+ 2-Reu
= 0,
on
S2
109
has a solution. Below we shall always assume R > 0 and discuss the solvability of the equation (1.3). In section 2, we recall a necessary condition, say, Kazdan-Warner condition, and give a necessary and sufficient condition (Theorem 2.1). Section 3 is about the existence of solutions of (1.3), including a degree theoryTheorem 3.1 or Theorem 3.2, which is shown to improve some important results previous (Corollary 3.1-Corollary 3.3). The main steps for proofs of these two theorems will be given in section 5. And we shall collect in section 4 some basic inequalities and estimates previous that plays a fundamental role in the study of the Nirenberg problem.
§2. K a z d a n - W a r n e r condition The solvability of (1.3) can be reduced to a variational problem: a critical point of the functional (2.0)
J{u) = \f | v « | 2 - 8 7 r l o g / ReudA, 2 2 Js Js2
Vu €
ff1^2),
yields a solution to (1.3). This functional is bounded from below, however it has no minimum except R =const. and is lacking of compactness. For the solvability, various kinds of sufficient conditions are given, which we are going to talk about in next section in more detail. First of all, it is known that it is not always solvable and there is a necessary condition, Kazdan-Warner condition, which says KW^0 where (2.1)
KW—
{ueH^S2)
: / sjR- v ^ i e" dA = 0, i = 1,2,3}. Js2 This gives rise to many examples for which (1.3) has no solution. For example, if the function R is taken to be R = x\ + c with c > 1 a constant, / 2 \jR-sjxxeu Js
dA=
I 2 | ya:i|2 Js
eudA>0
110
for any u G H1(S2), then the set KW is empty and the equation (1.3) has no solution by Kazdan-Warner condition. Precisely speaking, they proved ([19]) the following K - W Theorem. If u solves (1.3), then u G KW. Proof. For u G H1, set Vk = y u ' V 1 * (k — 1,2,3). A direct calculus yields / y u • \7vkdA Js2 / V u • V ( V U • \/xk)dA
-
V(l V "I 2 ) ' S/XkdA - / 2 | v Js2 Js A / 2 | y u | 2 ( - A i t -2xfc)d^ Js 'S2 0
=
h
= =
u\2xkdA
and /
ReuvkdA
2
h =
/
=
-
=
2
R v (e u ) • yx fc cL4 / y i ? • yz f c e u cL4 - / i? e u A a;fccL4 .As2 Vs=
Vs2
R euxkdA
- / 2 y i ? • yz fc e u cL4 ^s
since — Axk = 2 Xk
k = 1,2,3.
It follows by the definition of J that for k = 1,2,3, < dJ(u), ufc > =
/ y u • yufcdA + 2 / ufccL4 Js2 Js2
/ y u • \7xkdA Js2 + -Mj Reu
[ s
J 2
j I Reu
\/R-\/xkeudA
^^— / R / Reu Js2
/ R Js2
euxkdA
euvkdA
111
and < dJ(u), xk > = / V " • \jxkdA Js2 since
L is2
-r /
Reu
/ R Js2
euxkdA
xkdA — 0.
Thus we get (2.2)
< dJ(u), vk>=2<
dJ{u), xk > +—
/ u E>„« Re
/
V-R • V ^ e u dA
JS2 Js2
for any u £ Hl. Now let u be a solution, then < dJ(u), vk >= 0; < dJ(u), xk >— 0, which together with (2.2) yield that
L is2
V # • S?xkeudA = 0
A; = 1,2,3.
This means u £ KW, finishing the proof.
•
Whether is Kazdan-Warner condition sufficient? In 1987 BourguignonEzin [2] found a slightly more general condition than Kazdan-Warner condition obstructing the solvability of (1.3) and gave an example of R that satisfies Kazdan-Wamer condition, but not theirs. In 1995 Chen-Li ([10], [11]) obtained some new nonexistence results where R is axi-symmetric and satisfies the Kazdan-Warner condition. All of these illustrates that the Kazdan-Warner condition is not sufficient. Then a problem is how far Kazdan-Warner condition being from sufficiency. Below we suitably strengthen Kazdan-Warner condition, so as to be both necessary and sufficient. Let Ho be the Hilbert space flo^uefl^2):
/ u = Q}. Js2
112
Denote B = {x £ R3 : x\ + x\ + x\ < 1}. Like Chen-Ding [8] and the others, we introduce the center-of-mass map P : H1 ->• B by (2.3)
P(u)=
[ xeu^dx Js2
/ [ e<xUx, I Js2
VuGH1.
For a £ B, set (2.4)
Ma = {ueH0:
P(u) = a}.
It is not difficult to prove that Ma is a submanifold of the natural Finsler structure in H0, by showing that the differential operators {a!Pi(u)}i
Ka = {u£Ma:
dJa(u) = 0}.
Finally let
(2.6)
S := (J
Ka.
Theorem 2.1. The equation (1.3) is solvable if and only if (2.7)
SflKW#0
where the sets KW and S are defined by (2.1) and (2.6) respectively. Proof. Let u be a solution. By K-W theorem, u €KW. And by GaussBonnet theorem, /
Js2
ReudA = 87r, then u is a critical point of J. It is
naturally critical for the restriction Ja with o = P(u), then u e Ka C S. Thus u £ S fl KW. The necessity is shown. For sufficiency, let u £ SnKW. Then u £ Ka for some a £ B, a critical point of the functional J a . Below it will be shown from u £KW that u must be also critical for J. Hence u + c solves (1.3) for some constant c. The solvability is obtained. Indeed we first characterize the tangent space of M n at u, i.e., Tu(Ma). By P(u) = a, / eu(x - a) vdA < dP(u), v > = ^ — / eudA Js2
W £ Ho,
113
we then have (2.8)
Tu(Ma)
= {veH0:
I eu{x - a) vdA = 0}. Js2
Denote (2.9)
T
H = /2 Js
eU x
i i ~ a,i){xj - a,j)
(i,j = 1,2,3),
then T := (^3)3x3 is a positively definite matrix. We denote T For v eH^S2), set
l
= (T 8J ).
v := —- I vdA\ ±* Jap 's2 (2.10)
ai=
2
eu{xi-ai)v
(i = 1,2,3);
Js
pi = ^ajTi'
(2.11)
(* = 1.2,3);
and set {; : = (v -v)
(2.12)
We claim that v 6 Tu(Ma).
-y^^PiXj.
Indeed, / vdA = - ^ Pi / %idA = 0, i.e. Js2 ~[ Js2
v € Ho, and for j = 1,2,3, / eu(xj — a,j)v = / eu(a;j — aj) (u — v) — / J $ 3 is2 Js 2 [ j=1 =
/ 2 eu(a;j - a , > ~Y]Pi Vs ~J
Js
2
eu(a;i - oi)(x:(- - a.,)
3
= ay-£>!!,• = 0 »=i
by using P(u) = a and (2.9)-(2.11), thus u € Tu(Ma) in terms of (2.8), and the assertion is obtained. Now from dJa(u) = 0 we have
VwGff 1
114
where v is defined by (2.12). Choose v = vk = V w " \7xk {k = 1,2,3) and aki is determined by (2.10) with v = vk, i.e., (2.13)
aki = /
eu(Xi - en) vk dA
Vfc.i = 1,2,3,
and 3
Pki = Y,ai
(1.14)
(*,» = 1,2,3),
then we have 3
vk:=(vk-vk)-^2PkiXi
eTu{Ma)
(k = 1,2,3).
2=1
It follows that (2.15)
< dJ(u), vk >= 0
i.e. 3
(2.16)
vk-vk>=Y^Pki
Xi>
Vfc = 1,2,3.
»=i
According to the identity (2.2) (in the proof of K-W Theorem) and the fact that < dJ(u), c > = 0 for any constant c, we see the left hand side of (2.16) equals 2 < dJ(u), xk > H— / V-R • \ Reu Js2
\/xkeudA,
equaling 2 < dJ(u), xk > by u £ KW. Thus (2.16) becomes 3
(2.17)
2
xk >=YjPki
Xi
>,
t=i
where (3ki is defined in terms of aki by (2.13) and (2.14).
Vfc=l,2,3
115
Carefully computing aki in (2.13), we have that for k,i — 1,2,3, Q-ki =
eu(%i - flj) V w •
/
S/xkdA
(xi - a) V (eu) • S7xkdA
s2 / /
eu v Xi- \/xkdA
- /
eu\7Xi-yxkdA
+2
eu(xi - a*) A eu{xt-
e" V xi- VxkdA + 2 euVXi5
xkdA,
a,i)xkdA
eu{xi - a,i)(xk - afc)d,4
VxkdA + 2Tki
2
by making use of — A xk = 2 xk and P(u) = a and (2.9). Denote Gki=
e" V Xi • \jxkdA,
k,i = 1,2,3,
and denote G := (Gki)3x3, which is a positively definite matrix. Then we have the expression: (aki) = — G + 2T, and (Pki) — (aki) T - 1 = -GT-X + 27 by (2.14). Substituting it into (2.17) and noticing that det \G\ det \T\~l > 0, we obtain
VuGH 1 ,
with u is defined by (2.12), imply that < dJ(u), v>=0 Thus dJ(u) = 0. The sufficiency is shown. The proof is finished.
v G H1
Q
Remark 2.2. It is good that the restricted functional Ja, for each parameter a £ B, is compact, satisfying the Palais-Smale condition. Hence there is no any difficulty in analysis to determine the critical set Ka and so S. At least, Ja has a minimum ua (Va & B), and S contains all of these minima. By carefully investigating these minima and the set KW, we shall arrive in the next section at a degree theory for the solvability.
116
The point of Theorem 2.1 is to turn the problem for the bad functional J (lacking compactness) into that for the good functionals Ja (being compact) (a e B) through Kazdan-Warner set KW. In other words, the set KW bears the responsibility for the lacking of compactness of the problem. Clearly, an investigation of the set KW is also helpful in understanding the problem. Roughly speaking, the bigger KW is, the more possible the equation (1.3) admits a solution. For example, when R is a constant, the set KW is biggest, being the whole H1, then E n KW=E is not empty, leading to the solvability, a well known fact. Another example is the case where R is even. In this case, we observe that {even functions} CKW. Indeed, because of the symmetry, it suffices to show (2.18)
/ v # - VZ3 e" dA = 0 Js2
for even functions u. Let (8,(/>), —TT/2 < 8 < ir/2, —ir <
s/R-\/x3eu
0) = u(-0,
U{6,
is an odd function of 0, then dA=
f J—x/2
cos2 6d0 f
i?0(0,)e u(M) # = 0.
J-iT
This verifies (2.18) Thus, the set S n KW contains S D {even functions}. Moreover, it is clear that E n {even functions} ^ 0 since R is even. We see E n KW / 0, and the equation (1.3) is solvable by Theorem 2.1. This is just the famous result due to Moser ([21]).
§3. Existence of solutions In recent almost thirty years, there have been a lot of works about the existence of solutions to (1.3) and various kinds of sufficient conditions have been given.
117
The pioneer work is due to Moser who proved the existence in 1973 for R being even(as we mentioned in last section) by minimizing the functional J in the space of even functions. Later Hong([15]), Xu-Yang([25]) considered R as a axi-symmetric function. And Chen-Ding([8]), Ji([17]) studied the case where R is general symmetric, say G-symmetric, R(Gx) = R(x), Va; £ S2, with G an orthogonal transformation group on S 2 . In these special cases, some other conditions are imposed for the solvability. When R is not necessarily symmetric, there are also many studies([3]-[7], [9], [12], [14], [16]). Since the functional J has no minima in general, people made efforts to look for minimax type of solutions, such as, by establishing a mountainpass-lemma, Morse theory, and other minimax schemes. The following result is due to Chang-Yang ([5],[6]): Chang-Yang Theorem. Suppose R has only isolated non-degenerate critical points and in addition satisfies (3.1)
| v -Rl + I A H| ^ 0.
If £y6S_(-i)ind(^i, then the equation (1.3) has a solution, where 5_ := { y e S 2 : vR(y)
= 0, AR(y) < 0}.
This result was proved, as a main result, first in [5] by using a delicate minimax procedure and analyzing where the minimax sequence fails to converge, later proved in [4] and [14] by establishing a Morse theory for the functional J. Later in [3] and [7], the existence of solutions to (1.3) was shown when a map G, associated to the function R, has non-zero degree. Here the map G was defined using the action of the conformal group of S2, and has an integral expression. They considered the following set of conformal transformations of S2. Given P e S2, t > 1, using y as the stereographic projection from S2 — {P} (where P is the north pole) to the equatorial plane 2/i,2/2- Let (j)pj be the conformal map of S2 given by (f)p,t(y) = ty. The totality of all such transformations comprise a set which is diffeomorphic to
118
the unit ball B, with the identity transformation identified with the origin in B and (j>ptt o ( "7 )P — p G B in general. They constructed the map G : B -> R3 by setting (3.2)
G(P,t)=
[
(Ro(f>Pt).xdx.
It was motivated by Kazdan-Warner condition since the condition than expresses the fact that the variation of J vanishes along the conformal transformations. They proved that if (3.1) holds, and if deg(G,B,0) ^ 0, then the equation (1.3) has a solution. As an application of it, Chang-Yang Theorem is derived again in [3]. Enlightened by Theorem 2.1, a necessary and sufficient condition to the solvability of (1.3), below we are going to investigate the sets KW and S. In fact, we shall not consider the whole X, but only a subset of it, the set of minima of the functional Ja for all a G B. Notice the fact that (3.3)
J(u + c) = J(u),
P(u + c) = P{u)
for any constant c. For a G B, we set (3.4)
Ma = {u + c : u e Ma,
c is any constant},
and denote by ua 6 Ma a minimizer of Ja satisfying / e"° = 4ir, then define (3.5)
F{a) = -!- /
v # • Vz e"° da.
47r y S 2
This way we obtain a multi-valued map F: B —> R3. Now the problem is to solve the equation (3.6)
F{a) = 0
a G B,
according to Theorem 2.1. For this purpose, we shall show that the map F has the Brouwer degree at zero point being the same as the following map (3.7)
G(x) = S7R{x) • sjx - AR(x)x,
xeS2,
119 provided that F is continuous in the interior of B (Lemma 5.2), then verify the continuity when the function R is close to 2 (Lemma 5.5). That means the equation (3.6) has a solution if deg(G, B, 0) ^ 0, and the solvability of (1.3) is obtained in this case. Moreover, based on some estimates for the solutions to (1.3) in [3], and some further calculus for degree, we remove the restriction that R is close to 2 and arrive at the following T h e o r e m 3.1 Suppose R satisfies | v R\ + I A R\ ? 0. If deg(G, B, 0) ^ 0, so the equation (1.3) has a solution, where the map G is denned by (3.7). Here we remark that, very obviously the map G does not vanish on 5 2 that is the boundary of B under the condition (3.1), so the Brouwer degree of G at 0 is well-defined. Below we change Theorem 3.1 into a simpler form Theorem 3.2. Let N = (0,0,1) be the north pole. Identify S2\{N} with the equatorial plane 2/1,2/2 via the stereographic projection: y : S2\{N} -* R2 xeS2,
(3.8)
1/1 = : ^ - ; V2 = j ^ - , 1-X3 l-x3 with its inverse transformation ,,. (3.9)
22/! Xi
=
• , , , , ,2
1 + I2/I
22/2 %2 =
,
,
I 122
1 + I2/I
|2/|2-1
X
3
-
•,
, |
12'
1+ bl2
and y = 00 corresponding to N. Then (3.10)
50 = dx\ + dx\ + dx\ = ———(dyl
+ dyl).
Now R and AR can be also viewed as functions of y in R2, and V ^ be viewed as a map from R2 to R2. Without loss of generality, we assume S7R(N) = 0,
120
which guarantees that as y —> oo,
Introduce G* : R2 ->• it 3 by (3.12)
G*(i/):=(vii(l/),-Ai2(y))
w
V 2
dyi'
2
Vy G i? 2 ,
9^'
^7
Because of (3.11), G*(y) ->• (0, 0, - A i?(7V))
as j/ -> oo,
where AR(N) ^ 0(by assumption (3.1) and V-R(N) = 0). We see that G* is a continuous map from S2 — R2 U {oo} into i? 3 \{0}. Thus ygm- continuously maps S2 into S2. Using Theorem 3.1, we can prove the following T h e o r e m 3.2. Suppose (3.1) holds. In the case AR(N) < 0, if deg(|^rr) ^ 1, then (1.3) has a solution; in the case AR(N) > 0, if" deg(j^rj) ^ —1, then (1.3) has a solution. Here G* is defined by (3.12). We shall give the main idea of proofs for Theorem 3.1 or Theorem 3.2 in section 5. In Theorem 3.1 or Theorem 3.2, the map G or G* has very simple expression: explicitly and simply depends on R, in fact, only on \/R and AR. It is not hard to calculus their degree. In the following, we give a few corollaries. Corollary 3 . 1 . Suppose (3.1) holds. If AR(x) A R(-x)
- yR{x)
• \/R(-x)
> 0 Vx £ S2,
then the equation (1.3) has a solution. Proof. Set Rt(x) = R(x) + tR(-x)
Vz e S2, te [0,1].
121
The corresponding map G = Gt {t e [0,1]), defined by (3.7) with R = Rt, satisfies |G ( (x)| 2
= | V R{x) - t v R{~x)\2 + (AR{x) + t A R{-x))2 > | V R(x)\2 + ( A E ( i ) ) 2 + 2t(AR(x) A R(-x) - \7R{x) • >\yR(x)\2 + (AR(x))2 >0
\/R(-x)),
by (3.1) and the condition AR(x) A R(-x)
- \/R{x) • yR(-x)
> 0.
So Gt has the degree independent of t by the homotopy invariant property. Observe that Ri(x) = R(x) + R(—x) is an even function and the corresponding map G\ is odd, hence has an odd degree by Borsuk theorem. We see that the map G = Go, corresponding to R, has the odd degree. Application of Theorem 3.1 immediately yields the conclusion. The proof is finished. When R is even, i.e. Moser's case, obviously AR(x) A R(-x)
- \?R{x) • \7R(-x)
= \ y R(x)\2 +
(AR(x))2,
being not negative, Va; G S2. Thus Corollary 3.1 is an extension for the even function cases. Corollary 3.2. Suppose (3.1) holds. If R only has non-degenerate critical points in {AR > 0}(or in {AR < 0}), and £ y e 5 + ( - l ) i n d ( y ) ± 1 ( or S y e S _ ( - l ) i n d W jt 1), then the equation (1.3) has a solution. Here 5+ := {yeS2:
VR(y)
= 0, AR(y) > 0}
S- := {yeS2:
yR(y)
= 0, AR(y) < 0}.
Corollary 3.2 extends Chang-Yang Theorem in which all critical points of R are assumed to be non-degenerate, so R is only permitted to have
122
isolated, finite number of critical points. In Corollary 3.2, R may have infinitely many critical points in {AR < 0}(or in {AR > 0}). Proof of Corollary 3.2. Denote by (u\, 112, U3) S R3, u\ + u\ + u\ = 1, the image of the map j^nrr. Take {111,112) as the coordinates at the north pole (0, 0, 1) and the south pole (0, 0, -1). By definition (3.12) of G*, \G*(y)\(Ul,u2)(y)
=vR{y) _ (i+\v\2 dR i+|y[2 dR\ ~ \ 2 dVl > 2 dy2 J •
Then it is easily seen that, at the point y satisfying S?R{y) = 0 and y ^ 00, dui
(3.13)
sign
du-2 dyi
sign
R.vwi ;
Ry R V2V1
Ry
= sign\D2 R(y)\
where D2R is the Hessian of R. Second! observe that for the map ^77, the inverse of the south pole Secondly, (0, 0, -1) is S+ = {y € S2 : vR(y) = 0, AR(y) > 0}, while the inverse of the north pole (0,0,1) is
S_ = { i / e S 2 : v % ) = 0,A%)<0}. If we suppose in S+ the function R is non-degenerate, the determinant in (3.13) is not vanishing with the sign equaling sign (-l) 11101 ^) for y ^ 00, y £ S+. Choose a maximum of R as N, i.e. R(N) = maxs2 R, then Ai?(JV) < 0 and S+ does not contain 00. It follows that the determinant in (3.13) is not vanishing for y 6 S+, then the south pole (0,0, —1) is regular. Moreover, (3.14)
G* d e g ( — - ) = E y e s + sign I" I
du\
8m
01X2
dyi 9U2 9j/2
= SyeS+(-l)ind^).
On the contrary if we suppose R in S- is non-degenerate, we choose a minimum as N, i.e. R(N) = mins2 R, then AR(N) > 0, implying 5_ does not contain 00. Similarly as above we see that the north pole (0,0,1) is regular. Notice that, if (ei,e 2 ) is an orienting basis for R2, an orienting
123
basis for S2 at the south pole (0,0, - 1 ) is (ei,e2), while at the north pole (0,0,1) (ei,— e 2 ) is orienting. So (3.15)
d e g ( - ^ r ) = - S y e s _ sign
dui 9yi du2 9yi
du\ dy2 du2 9y2
_-,und(y) = -Ss6S_(-l)
With (3.14) and (3.15), we apply Theorem 3.1 and immediately obtain the conclusion. The proof is finished. Let us go to the last corollary. Denote (3.16)
0 := {y € S2 : AR(y) > 0}.
Denote by g* the restriction of gradient of R on the boundary <9ft, i.e.,
Since AR = 0 on dfl, by (3.1) g* is a map into -R 2 \{0}, thus ^ where S 1 is the equator S1 = {u € R3 : |u| = 1, U3 = 0}.
: dSl -> S1,
Corollary 3.3. Suppose (3.1) holds. If A T extends to a map: Cl -» S1 (or to a map: S2\Cl -> S 1 ), then the equation (1.3) has a solution. Proof. Let /i : f) —> S1 be a continuous extension of Arr. Since both I^TT (with G* defined by (3.12)) and h map ft into the lower hemisphere {«3 < 0} with the same boundary value T^T, and since the hemisphere can be viewed as R2, there is homotopy, i.e. a continuous H : [0,1] x fi —> S2, satisfying ff(0,-) = | | ^ ; and
H(1, •) = /»,
9* G* H{t,-)\ea = | ^ - | = | ^ j |an-
By defining
in[0,l]x52\n,
H=~
we get a continuous H : [0,1] x S2 —» S2, which is an extension of H, satisfying C* U H(0,-)=
iG*r
124
so T^prr is homotopic to H(l, •) : S2 —> S2 which is an extension of h. Notice that T^rr maps S2\Cl into the upper hemisphere, then F(l,S2)c{u3>0} and hence H(l, •) has the degree 0. Thus deg(j^7j) = 0. In another case when A T extends to a map h : S2\Cl —> S1, a similar argument as above shows that Mpr\ is homotopic to a continuous h : S2 —>• S2 with h{S2) C {u3 < 0}, implying deg(jg^j) = 0 too. By Theorem 3.2 we finish the proof. The assumption in Corollary 3.3 that -PTT extends to Cl (or to S2\Cl) is equivalent to deg(j^|9n') = 0 10 I where Cl' is any connected components in Cl (or in S2\Cl), according to a fundamental theorem in topology, in the case where A is a manifold. For example, if R is an axi-symmetric function, for any annulus type connected components Cl' in Cl (or in S2\Cl), the degree of Arr\dQ' is vanishing. To the best of my knowledge, there seems no any counterpart of Corollary 3.3 in the past.
§4. Preliminaries In this section, we list some basic inequalities and estimates about the functional J and solutions of equation (1.3). First introduce some notations. For every a £ B, define a function
Mx)
= l g
° (CoS2l/x2Sin2l)2
XGS2
where 6 = d(x, r M the distance on (5 2 , q0) between x and -A-, and A = v a |a| ' \a\ Aa £ (0,1] is uniquely determined by the equation P((f>a) = a.
125
Set (4.2)
I(u) = l [ \yu\2dA 2 Js2
u^H^S2).
+ 2 [ udA, Js2
For ( 6 S 2 , ( 5 > 0 , denote (4.3)
Cc^ueff1^2);
Q(u) = (,d(u) = 8}
where (4.4)
Q(«) = P(«)/|P(u)|,
d(u) = | Q ( t t ) - P ( « ) |
if
P(u)^0.
Lemma 4.1 ([15], [23]). (1). For a € B there exists a unique conformal transformation Ta : 2 S -J- S 2 such that dy = e^adx
if
y = Ttt(ar).
(2). For a E B, <j)a has the properties: / e^° = 47r;
and
<^>a € Ma.
(3). We have /" eu < 4TT exp J(u)
tf1^2),
Vu €
with the equality if and only if u = <pa for a £ B. Lemma 4.2 ([8]). Let {ui}, i = 1,2, • • •, be a sequence satisfying | P ( U J ) | < 5 < 1 and J(tii) < c < +oo, then there exists a constant C, depending only on 1 — S and c, such that f \yUi\2dA
Vi = 1,2,3- ••.
Lemma 4.3. (Corollary 5.1 of [5]). Suppose u £ C^ts with i"(u) = 0(<5/3) for some y3 > 0 and 5 sufficiently small, / 2 eudA = An. Then for every function h e
C2(S2),
Js
^ J2 /ieudA = MO + ^ Aft(C)5+ 0(5/{- logS)).
126
Lemma 4.4. (Theorem 2(a) of [3]). Suppose R is a smooth function on 5 2 satisfying (1.3) with 0 < m < R < M, then there exists a constant C\ > 0, depending on m,M and min{| A R(x)\ : \jR{x) = 0}, such that for all solutions u to (1.3), |u| < C\. For proving our Theorem 3.1, the above Lemma 4.4 can not be directly applied. We need the following Lemma 4.5, a modified estimate. Let X the Banach space X = {u£C2'a(S2):
fu = 0},
and Ls : X -» X defined by (4.5)
Lsu = A " 1 (2 -
8?r
Rs eu)
Vwel
u
Rse where (4.6)
V 0 < s < 1, x G 5 2 .
R,(x) = 2 - 2s + sR(x)
Lemma 4.5. Suppose (3.1) holds. Then the set S = {ueX
: u - Lsu = 0, 0 < s < 1}
is bounded in X. Proof. If u £ S, for some s £ (0,1] we have (4.7),
-Au + 2-
&n
u
Rseu
=0
onS2,
I Rse
then, for a constant cu, v := u + cu satisfies (4.8),
- A w + 2 - t f , ev = 0
Since / u — 0, we see c u = ^
onS2.
v, the bound of u implies the bound of u,
thus it suffices to show that the set of all solutions to (4.8), (0 < s < 1) is bounded in C2,a.
127
Above Lemma 4.4 says that all solutions u to (1.3) have the estimate: ||u||c2,c < C where the constant C depends only on mini?, maxi?, and min{| A R(p)\ : S7R{p) = 0}. If we directly apply Lemma 4.4, it can only yield ||w||C2,<» < Cs where Cs depends on s > 0. However, by carefully observing the proof of it in [3] and noticing that our function Rs has the special property: \/Rs = s y R, we can find that all solutions to (4.8)5 have an uniform C 2 ' a -estimate independent of 0 < s < 1. Indeed, on the contrary suppose that there are Sk > 0; Vk satisfying (4.8)Sfc, but H^feilc° ->• oo as A; -^ oo. The same as in [3](pp.209), we get a new sequence Wk satisfying ||t«fe||cri = o(l) (as k -> oo) and
| v ( f l . * ° l k ) • s/x ew" = 0
(4.9)
where tpk = Tah with a^ = (1 - \k)P for some fixed point P G S2 and Afc G [0,1); Afc -> 1. By ^RSk = SkSJ R, we see (4.9) becomes (4.10)
v{Ro1>k).s?xev"-=0.
/
It is exactly the formula (3.10) in [3] where it was shown that their (3.10) implies V-R(-P) = 0 and AR{P) = 0. So above (4.10) implies vR(P) = AR(P) = 0, contradicting to the assumption (3.1). The proof is finished.
§5. Main steps of proofs In this section we give the main steps of proving Theorem 3.1 and Theorem 3.2. We omit the details and proofs of all lemmas. Main idea for proof of Theorem 3.2. Using y as the stereographic projection (3.8) and (3.9) from in S2\{N} to the equatorial plane 2/1,2/2Writing
x=\
x2
x* J
;
G*(y)=\
M™
\ -AR
J
(x\,X2,Xz)
128
Denote A
yy>
-
\
2
9j» '
2
X
dy2 >
)
( 1 + 2/1 — 2/i -2j/i2/2 -22/12/2 1 + y\ ~ 2/2 V 2^ 2y2
= i+fcp
Then by definition of G and G*, it follows (5.1)
Vx 6 S2;
G(x)=A(y)G*(y)
y 6 R2.
Here the matrix A(y), y £ R2, is easily verified to be orthogonal with A2 =id. It should be mentioned that in (5.1), the matrix A is not continuous on S2 (discontinuous at the infinity, the north pole N), however it is easy to see that if a continuous / : S2 -¥ S2 maps the north pole N to N or to the south pole (0,0,-1), then Af : S2 -> S2 is a continuous map. That is a reason why we always assume \?R(N) — 0. Let k be an integer, and introduce Ik : S2 —>• 5 2 by
(
2r cos fc(/> 2r sin k<j> r2-l
where (r, <j>) being the polar coordinates: yi—r cos (j>, 2/2 = r sin (/>,
Vr > 0, 0 < <j> < 2ir.
Assume deg(T^rr) — m. We prove the following (1) in the case AR(N) < 0, deg(G, B, 0) = (2)in the case AR(N)
deg(AIm);
> 0,
deg(G, B, 0) = -deg(Afi),
with / = - m ;
(3)for any integer, it is true deg^J*) = 1 - k.
129
With these facts (l)-(3), we obtain deg(G, B, 0) = 1 - deg( | | 1 )
if A R(N) < 0;
deg(G, B, 0) = - 1 - d e g ( | | ^ )
if A R(N) > 0.
Then Theorem 3.2 is obtained by Theorem 3.1. Main idea for proof of Theorem 3.1. Step 1. Calculus the degree of the map F, under the assumption of the continuity. Here F is defined by (3.5), i.e. F(a) = — / V-R • V z e"° dx 4TT JS2
aeB
with ua G Ma being a minimizer of J a satisfying / 2 e"° = 4-7T. is In this step, a key point is the following Lemma 5.1. Let / € C3(S2). < x, \jf{x)
Then Vz G S 2 ,
• y z > = 0;
< x, A ( V / ( z ) • v z ) > = - 2
A
/(a?)-
Here < •, • > is the inner product of R3. Applying the asymptotic formula Lemma 4.3 we first show that the map F has the expression: when 5 > 0 sufficiently small and a G B with \a\ = 1 — 6, F(a)
=
4^r /
v i ?
'v x
= V-R(a;) • V
x
+\
e
"° A
^
(VR(X) • Va;)<5 + o((5).
Secondly, using Lemma 5.1 and the condition (3.1), we can construct a homotopy H : [0,1] x S2 such that H never vanishes and satisfies H(0,-)=F;
130
H(l,x)
= V-R- S/x-
xeS2.
ARx
Then it proves the following Lemma 5.2. Suppose (3.1) holds. There exists e > 0, depending on ||.R||C2, such that d e g ( F , B r , 0 ) = deg(G, B, 0)
when 1 - e < r < 1,
provided that the correspondence a H-> ua is a continuous map from Br to H1. Here G is defined by (3.7). Step 2. Establish an inequality of Poincare type in some subspaces of H1. It will play a fundamental role in later step 3 and step 4. We recall that, for a € B, ua € Ma is a minimizer of Ja satisfying /
Js2
eUa = A-K, it also depends on the function R, and TUaMa denote the
tangent space of Ma, at ua, i.e. the set of functions v € H1 satisfying / eUa(x - a)v = 0) (see (2.8)). Let Ta : 5 2 -»• S2 the unique conformal transformation determined by Lemma 4.1(1). We observe, from Lemma 4.1(3), that ua =
By using Lemma 5.3 and some careful calculus, we can prove Lemma 5.4. Let 0 < r$ < 1. There exist 5 > 0 and p, > 2 such that, when |i? — 2| c o < 5, there holds
/ l v < f > M / l«°77T
is 2
Js2
131
for every a G BTQ and v G TUaMa. Step 3. Show the continuity for the correspondence a 4 « 0 when R is close to the constant 2. It suffices to show the uniqueness of minimizers of Ja subject to / 2 e"° = Js An. As we mentioned in step 2, obviously it is true when R = 2. For R close to 2, we can prove the uniqueness based on Lemma 5.3 and Lemma 5.4, that is the following L e m m a 5.5. Given e G (0,1), there exists 6 > 0 such that, when \R — 2|c?o < S, for every a G Bi_ e , the functional Ja has minimum at unique ua G Ma with / eUa = 4ir. Combining Lemma 5.2 and Lemma 5.5, we have proved Theorem 3.1 in the case R is close to 2, by Theorem 2.1. Moreover, in order to remove the restriction, we need to pay attention to the Leray-Schauder degree of a map related to equation (1.3), still in the case R close to 2. It is what we will do in the next step. Step 4. Let X the Banach space X = {u £ C2'a(S2)
: / u = 0}, and Ls : X ->
X defined by (4.5) and (4.6), i.e. Lsu = A " 1 (2 -
8?r
/Rse
Rs eu)
V«£l
u
where Rs(x) = 2 - 2s + sR(x)
V0 <s < 1, x G S2.
We shall compute the Leray-Schauder degree deg(i — Ls, fi, 0) with s close to 0, where the set ft c I is taken to be an open bounded set that contains S, the set of zero points of (I — Ls), Vs G (0,1], and S D dfi. = 0 ( by Lemma 4.5). In fact, we wish to prove that for some sufficiently small si > 0, (5.2)
deg(7 - LS1, n, 0) = ± deg(G, B, 0).
132
For this purpose, we first prove L e m m a 5.6. Given M > 0, there exists 6 > 0 such that, when \R 2|co < S, any solutions u of (1.3) satisfying ||u||c2,<» < M, must be the unique minimizer of Ja subject to / Reu = 8ir, where a = P(u). Js2 ^From this lemma we can see that, when s close to 0 (implying Rs close to 2), any zero point u of the operator (I — Ls) must be the unique minimizer of Ja subject to / Rseu = 8ir where a = P(u). Based on this knowledge about the zero points, and based on Lemma 5.3 and Lemma 5.4, we can construct a series of homotopy H(t, •), t > 0, that never vanishes on the boundary Oil and deform the restriction (/ — Lai)\aa, for some sufficiently small si > 0, to a map like G x I: S2 x U, with U an open set of a subspace in X of codimension 3. This leads to the above formula (5.2). Step 5. Complete the proof of Theorem 3.1. The Leray-Schauder degree deg((7 — Ls), CI, 0) is well defined and independent of s € [s\, 1]). Then from last step we have deg((J - LJ,
n, 0) = ±deg(G, B, 0) # 0.
This means that the operator (I — Li) has a zero point, which produces a solution of the equation (1.3).
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P E R I O D I C M E A N CURVATURE A N D BEZIER CURVES
K. KENMOTSU Mathematical Institute, Tohoku University 980-8578 Sendai, Japan E-mail: kenmotsu@math. tohoku. ac.jp We show a recipe to get the periodic surface of revolution in the Euclidean three space such that its mean curvature is a given function.
1. Introduction It is well known that the profile curves of surfaces of revolution with constant mean curvature in R3 are constructed by the rollings of ellipses or hyperbola along the axis of rotation (Delaunay x ). In the recent paper by Kenmotsu 3 , he showed a new method to have such profile curves: In fact a circle with radius r produces one parameter family of surfaces of revolution with constant mean curvature l/2r. Moreover, generalizing this, he found all profile curves that make periodic surfaces of revolution and proved that these are provided by a class of Bezier curves. The purpose of this paper is to show the figures of these periodic profile curves and the commands of Mathematica which is a symbolic manipulation program by Wolfram. Moreover, the resemblance and the difference between those periodic surfaces and Delaunay surfaces are clarified. This paper3, extends the manuscript of my talk in the International symposium on differential geometry held in Shanghai on September 2001. 2. Periodic surfaces of revolution Let C = (x(s),y(s)) be a complete smooth curve parametrized by arc length, which generates the surface of revolution S in R3 as follows : S= {(x(s),y(s)
cos6,y(s) sin0) G R3 | s G R, 0 < 6 < 2TT} ,
where x-coordinate of R3 is the axis of rotation. C is called the profile curve of S. "Dedicated to Prof. Su Buchin on his centennial birthday
135
136
The surface of revolution S is called periodic if the coordinate function y(s) is periodic, that is, there is a positive number L such that y(s + L) — y(s) for all s E R. L is said to be the period of S. The mean curvature function of a periodic surface of revolution is periodic, but the converse does not hold in general, catenoid giving us the counter example. We have a necessary and sufficient condition for a continuous periodic function to be the mean curvature of some periodic surface of revolution by Kenmotsu 3 . To state this, given a continuous function H = H{s) on R, we put h(u) = = 22 / " H(s)ds, ueR. (1) Jo Jo Theorem 1. Let H(s) be a continuous periodic function on R with period L. Then, the function H(s) is the mean curvature of a periodic surface of revolution S with period L if and only if it satisfies the condition : Jo cos(h(u))du sin h(L)
=
JQ£ sin(h(u))du 1 - cos h(L) '
[
'
Convention. A denominator in Eq. (2) vanishes if and only if the corresponding numerator also does. We show now a recipe to get periodic profile curves : Let H(s) be a periodic function on R satisfying the criterion Eq. (2). Put F(s) = / sin(h(u))du, Jo
(3)
G{s) = [ cos(h{u))du .
(4)
Let c be the absolute value of the common ratio of Eq. (2) when some of the terms do not vanish and any positive number otherwise. Under these notations, we have a periodic profile curve C = (x(s;c),y(s;c)) due to Kenmotsu 3 as follows : y(s;c) = {(F(s)-c)2+G(s)2}i, V
;
Jo
(5)
{ ( F ( s ) - c ) 2 + G( s )2}l
V
'
This curve C is parametrized by arc length and passes through (0, c). When a periodic function H(s) trivially satisfies Eq. (2), we have a one parameter family of these curves with the same mean curvature which is given by {(x(s;c),y(s;c))eR2
| c > 0}.
137
When a periodic function H(s) non-trivially satisfies Eq. (2), for any c'(^ c) the curve (x(s;c'),y(s;c!)) generates a non-periodic surface of revolution with periodic mean curvature H(s). Hence, in this case, we have only one profile curve which is periodic.
Example 1. (Torus) Put ,. . /(S)
1/ , coss 1+ = 2l" 2^Ts)
\ •
-°°<«<°°-
(7)
It non-trivially satisfies Eq. (2) with L = 2TT and c = 1. In fact, this is the mean curvature of a torus in R3 whose profile curve is a circle and passes through (0,1). Since this function f(s) non-trivially satisfies the criterion Eq. (2), there is just one value c, in this case c = 1, such that the profile curve (x(s; l),y(s; 1)) through (0,1) is periodic, but for any other c (7^ 1) the profile curve (x(s;c),y(s;c)) is not. Figure 1 shows the curve {G(s), F(s)) made from f(s) and Figure 2 the profile curves (x(s;c),y(s;c)), -87r < s < 8n made from these G(s) and F(s) when c = 0.5, 1, 2.
Figure 1. The curve (G(s),F(s)) of a torus.
-2-1
1 2
-1-0.5 '
0.5
1
-4
-2
Figure 2. Profile curves with the same mean curvature We remark that the direct use of the command of Mathematica for the integration of Eq. (7) may not work well, because it is an improper integral. We integrate the Fourier expansion of Eq. (7) to prove Eq. (2). Let us consider the plane curve (G(s),F(s)), which is parametrized by arc length and has curvature 2H(s). This means that from the profile curve of a surface of revolution with mean curvature H(s), we have a plane curve
138
with curvature 2H(s). Reckziegel was first to remark that conversely, from a plane curve T with curvature k(s), a profile curve C whose surface of revolution has mean curvature k{s)/2 is constructed (See the first paper by Kenmotsu 2 .) Definition 1. (G(s),F(s)) is called the source curve of a surface of revolution S with mean curvature H(s). The curve (x(s;c),y(s;c)) is called to be associated with the source curve (G(s),F(s)). In section 3 we explain how to get periodic profile curves made from curvatures of closed plane curves and in section 4 we do how to make periodic profile curves from curvatures of some non-closed curves, in which Bezier curves are used. 3. Closed curves Let T be a twice continuously differentiable closed plane curve parametrized by arc length and L the length. The curvature of T, k(s), is considered as a periodic function on R with period L. We may assume that Y starts from the origin of (x, i/)-plane and is tangent to x-coordinte at the origin. It follows from the fundamental theorem of smooth curves theory that, on s € [0,L], we can write it as r = ( / cos ( /
k(t)dt\ du, J sin ( /
k(t)dt j du) .
(8)
With the famous theorem for the total curvature of closed plane curves, this formula implies that the function k(s)/2 trivially satisfies Eq. (2) with period L, because all terms in Eq. (2) vanish. Therefore, for any positive number c, there exists a periodic profile curve C(s; c) such that the resulting surface of revolution is periodic with period L, has the mean curvature k(s)/2 and passes through (0,c). We have seen that, a smooth closed curve with curvature k(s) provides the 1 parameter family of periodic profile curves {C(s;c) | 0 < c < oo}, where k(s)/2 is the mean curvature of resulting surfaces of revolution. This resembles Delaunay surfaces. The deformations of such periodic profile curves are now shown by examples : Example 2. (Circle) Let T be a circle with radius r centered at (0, r) e R2. The resulting {C(s;c) | 0 < c < oo} is the set of profile curves of De-
139
launay surfaces such that l/2r is the mean curvature of the corresponding surfaces of revolution. See Figure 3.
350 0.25
S.8D571.15 6 Figure 3. The deformation of profile curves made from circle We remark that this method is different from Delaunay 1.
E x a m p l e 3. (Ellipse) Let T be an ellipse such that one of the vertex is tangent to x-coordinate at the origin of (x, i/)-plane. The resulting 1parameter family of periodic profile curves resembles profile curves of Delaunay surfaces. See Figures 4 and 5.
5
2
10
4
15
6
Figure 4. The deformation of profile curves made from ellipse
140
Figure 5. The profile curves of surfaces of revolution associated with an ellipse. E x a m p l e 4. (Limagon) Let F be a Limagon given by (2acos[t] + 6)(cos[t],sin[t]),
— oo < t < oo,
where a and b are real constants. Figure 6 shows a Limagon, its curvature, the corresponding profile curve, and the resulting surface of revolution.
Figure 6. Lima^uii and the- associated bin face of n-volution Now we take a rose curve as F. Here is the command of Mathematica to draw the pictures of periodic surfaces of revolution associated with the rose curve. The rose curve is given by the command of a [ t _ , n _ , b _ ] := {2 * b * Cos[t] * Cos[n* t],2 * b * Cos[t] * Sim[n* t]};
141
Putting n := 2 ; b := 1/3 ;, Figure 7 shows the rose curve, which is given by the command of ParametricPlot[Evaluate[a[t, n, b]], {t, 0 , 2 * Pi}, AspectRatio —> Automatic]
Figure 7. The curve a[t, 2,1/3] Next we have the command to compute the curvature curve.
K.1[£]
of the rose
s'[t_] :=Simplify[(D[a[t,n,b],t].D[a[t,n,b],t]) 1 / a ] ; J[{pl_,p2_}]:={-p2,pl}; /cl[t_] :=D[a[tt,n,b],{tt,2}].J[D[a[tt,n,b],tt]]/s'[t] 3 /. tt -»• t ; nl[t] above is the curvature of the rose curve for the parameter t which is not arc length. Let s = s(t) be the transformation of parameters from t to arc length s. To get the function F(s) of Eq. (3), where s is arc length, we use the differential equation :
£ = - . ( / « i ( . ( . ) ) ! * ) ! . "(»)-»• We put k[t_] := Simplify[«l[tt] * s'[tt]]/.tt -> t ; Since this function is rather complicated to treat by Mathematica, we use the command of interpolation as follows : datal = Tableau, k[u]},{u, 0 , 2 * Pi + 0.1, 0.02}]//N ; K2 = Interpolation[datal] ; Now, we get F(s) = ff[s] and G(s) = gg[s] by /•tt
soil = NDSolve[{f [tt] = = Sin
/ 7o
«2[u]du *s'[tt],
142
/
g'[tt] = = Cos
re2[u]du *s'[tt], f[0] = = 0, g[0] = = 0},
{g,£},{tt,G,2*Pi}][[l]] ; ff[s_] := f[s]/. s o i l ; gg[s_] := g[s]/. s o i l ; Putting c = 1.5, we define y-coordinate of the profile curve through (0, c) by y[8_,c_]:=((ff[s]-c)a + gg[8]3)1/3; The next command computes ^-coordinate of the profile curve through (0,c). sol2 = NDSolve[{X'[t] — = I gg[*] * Sin
/
«;2[u]du
(ff[t] - c) * Cos
X[0] = = 0},{X},{t,0,2*Pi}][[l]] ;
/ K2[u]du J /y[t,c],
x[t_,c_] := X[t]/. so!2 ;
7 [ t - , c _ ] = {x[t,c],y[t,c]} ; Now, we have the figure of a profile curve through (0,1.5) which is made from the rose curve by : p i = ParametricPlot[7[t, c]//Evaluate, {t, 0,2 * Pi}, AspectRatio —»• Automatic, DisplayFunction —>• Identity] p2 = ParametricPlotpyjt, c] + {*y[2 * Pi, c][[l]], 0 } / / Evaluate, {t, 0,2 * Pi}, AspectRatio —J- Automatic, DisplayFunction -4- Identity] ; Show[pl,p2, DisplayFunction -*• §Display Function] Figure 8 shows the profile curve y[t, c], 0 < t < Ait, and the corresponding surface of revolution. By changing the constants n, b and c used here we find other profile curves.
8
">4/
Figure 8. The profile curve and the resulting surface of revolution associated with a Rose curve.
143
4. Bezier curves In this section we treat the nontrivial case of Eq. (2) such that there are non-zero terms in Eq. (2) . The Eq. (2) says that the coordinates of the end point of the curve (G(s),F(s)) are related to the tangent of the point by
2B. =
F
^
(9)
V F'(L) 1 - G'(L) ' We put po = (0,0), and let pi = (rcos8,rsin6),r ^ 0, be any point of R2 - {(0,0)}. Let us consider a C2 curve segment r 0 = (G(s),F(s)), 0 < s < L, parametrized by arc length which starts from po and terminates at Px such that the both tangent vectors at the end points are (G'(0), -^'(0)) = (1,0), and (G'(L),F'(L)) = (cos26,sin26). Moreover, we assume that the curvatures of the curve To at the both end points have the same values. We extend To by some isometries of R2 so as the resultant curve T is smooth, i.e., class C 2 , and complete. The curvature of T, k(s), is a continuous periodic function on R with period L and the function k(s)/2 satisfies Eq. (2) with period L. The source curves made above may not be closed and the periodic surfaces associated with the curves are isolated. That is, we have the special value c such that the resultant periodic curve starting at (0, c) is periodic, where the constant c is given by the common ratio in Eq. (2) and for any other c'(^ c), the profile curve starting at (0,c') is not periodic. Bezier curves are now used to find these source curves T0 from the given data of ends points. Let B(i), 0 < t < 1, be a Bezier curve with the control points bo,bi,...,b„ such that bo = (0,0),bi = (b, 0), ...,b n _i = (-6cos26' + cot 6, -bsin20 + 1), b„ - (cot 0,1), TT/4 < 6
B e r n s t e i n [i_Integer, n _ I n t e g e r , t_Integer] := Binomial [n, i] * t1 * (1 - t ) n _ i ; Bernstein [i_Integer, n_Integer]
144 = M a p [ B e r n s t e i n [ # , n , t ] & , R a n g e [0,n]] We define the Bezier curve with control points p t s by B e z i e r [ p t s _ , t _ ] := B e r n s t e i n [Length[pts] — l , t ] . p t s ; Let us take t h e following points, as a n example, p t s l = { { 0 , 0 } , { 0 . 4 , 0 } , {0., 1 } , { 1 , 0 . 6 } , { 1 , 1 } } ; By the command of B e z i e r [ p t s l , t ] , we have a parametric form of t h e Bezier curve with the 5 control points. T h e left hand side of figure 9 is obtained by N e e d s ["Utilities 'Filter-Options'"]; Options [BezierCurve] = {ShowPoints —¥ False}; BezierCurve [pts_,opts
] := Module [{t,ptsQ = S h o w P o i n t s / . { o p t s } / .
Options[BezierCurve]}, Show[Block[{§DisplayFunction = Identity}, ParametricPlot[Evaluate[Bezier[pts, t]], {t, 0 , 1 } , Evaluate[FilterOptions [Par ametricPlot, opts]], P l o t S t y l e ->• Thickness[0.001]]], Graphics[If [ptsQ, PointSize[0.02], Map[Point,pts]},{}]], FilterOptions[Graphics, opts],PlotRange -> AH]] ; b l = BezierCurve[ptsl, ShowPoints —> True, Frame —> True, A x e s —> False, P l o t R a n g e —>• {0,1.0}, AspectRatio —>• Automatic] In order to have the right hand side of Figure 9, we compute the curvature of the Bezier curve.
Figure 9. Bezier curve with 5 control points and the surface of revolution associated with the Bezier curve.
s l [ t _ ] :=Simplify[(D[Bezier[ptsl,tt],tt].D[Bezier[ptsl,tt],tt]) 1 / 2 ]/.
145 t t -»• t; J [ { p l _ , p 2 _ } ] := { - P 2 , p l } ; ic[t_] := D [ B e z i e r [ p t s l , t t ] , { t t , 2 } ] . J [ D [ B e z i e r [ p t s l , t t ] , t t ] ] / s l [ t ] 3 / . t t -» t ; d a t a 2 = T a b l e a u , /c[u] * sl[u]}, {u, 0 , 1 , 0.02}]//N ; K2 — I n t e r p o l a t i o n [ d a t a 2 ] ; We get F(s) = ff[s]
and G(s) = gg[s] by
s o i l = NDSolve[{f'[tt] = = Sin[Integrate[K2[u], {u, 0, tt}] * s'[tt], g'[tt]
= = Cos[Integrate[K2[u],{u,0,tt}] *s'[tt],f[0] = = 0,g[0] = = 0},
{g, f } , {tt, 0,1}])[[1]] ; ff[s_] := f [s]/. s o i l ;
gg[s_] := g[s]/. s o i l ;
Let us define y-coordinate of the profile curve through (0,1) by y[s_] := ((ff[s] - l ) 2 + ( g g H ) 2 ) 1 / 2 ; sol2 = NDSolve[{X'[tt] = = (gg[tt] * Sin[Integrate[/c2[u], {u,0, tt}] ~(ff[tt] - 1) * Cos[Integrate[/c2[u], {u, 0, tt}])/y[tt], X[0] = = 0}, {X}, {tt,0,l}])[[l]]/. t t - M ; ; we define x-coordinate of the profile curve by x[t_] := X[t]/. sol2 ;
c[t_] = {x[t], y[t]} ;
The profile curve is depicted by the commands of p i = P a r a m e t r i c P l o t [ c [ t ] / / E v a l u a t e , {t, 0 , 1 } , A s p e c t R a t i o —>• A u t o m a t i c ] p 2 = P a r a m e t r i c P l o t [ c [ t ] + {c[l][[l]], 0 } / / E v a l u a t e , {t, 0 , 1 } , A s p e c t R a t i o —• A u t o m a t i c ] ; S h o w [ p l , p 2 ] Finally, we have the commands to get the shape of the right hand side in Figure 9 as follows: p 3 = P a r a m e t r i c P l o t 3 D [ { x [ t ] , y [ t ] * C o s [ u ] , y [ t ] * Sin[u]}, { t , 0 , 1 } , {u, 0, 2 * Pi}], P l o t P o i n t s —¥ 50, A s p e c t R a t i o —> A u t o m a t i c ] ; p 4 = P a r a m e t r i c P l o t 3 D [ { x [ t ] + c[l][[l]], y[t] * Cos[u], y[t] * Sin[u]}, {t, 0 , 1 } , {u, 0,2 * P i } , P l o t P o i n t s -»• 50] Show[p3, p4, B o x e d —¥ False, A x e s —> False, A s p e c t R a t i o —¥ A u t o m a t i c ]
Acknowledgments This reserch was supported in part by the Grant-in-Aid for Scientific Reserch (No.12440012), Japan Society for the Promotion of Science 2001.
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References 1. C. Delaunay, Sur la surface de revolution dont la courbure moyenne est constante, J. Math. Pures. Appl. Ser. 1, 6(1841), 309-320. 2. K. Kenmotsu, Surfaces of revolution with prescribed mean curvature, Tohoku Math. J. 32(1980), 147-153. 3. K. Kenmotsu, Surfaces of revolution with periodic mean curvature, preprint, 2001.
Almost Complex Manifolds and a Differential Geometric Criterion for Hyperbolicity Shoshichi Kobayashi Department of Mathematics, University of California, Berkeley
0.
Introduction
The concept of almost complex structure was introduced by Ehresmann in 1947 [3], and the 1950s saw much activities in geometry of almost complex manifolds, including a major achievement by Newlander-Nirenberg [10] on integrability problem. The recent work of Gromov [5] and his school on symplectic geometry revived interest in almost complex structures; for every symplectic manifold admits a compatible almost complex structure, which can be profitably used to study the symplectic structure. The main difference between complex manifolds and almost complex manifolds is that a generic almost complex manifold does not admit, even locally, any nonconstant holomorphic functions. This difficulty coming from paucity of local holomorphic functions may be overcome by differential geometry, partial differential equations and hyperbolic complex analysis. We shall show here effectiveness of differential geometric methods in dealing with hyperbolicity. 1.
Almost complex manifolds
We recall that an almost complex structure on a real manifold M is an endomorphism J of the tangent bundle TM such that J o J = —I. Since the determinant det(—/) = (det J ) 2 must be positive, it follows that the dimension of M must be even if it is to admit an almost complex structure. We extend J complex linearly to the complexified tangent bundle TCM. The eigenvalues of J are ±i, and TCM decomposes (1.1)
TCM
= T'M + T"M,
where T'M and T"M correspond to the eigen values i and —i, respectively. We often identify TM with T'M by the map £eTM
->Z-iJ£e
147
T'M.
148
Under this identification, we have a natural orientation on M. If M is a complex manifold, then the decomposition (1.1) is naturally given, and setting J to be mutliplication by i on T'M and — i on T"M defines an almost complex structure on M. There are certain topological obstructions to the existence of almost complex structure. In general, the fc-th Pontrjagin class pk(M) of (the tangent bundle of) M is defined to be pk(M) = (-l)kc2k(TcM). If M admits an almost complex structure J, then c{TcM)
=
c{T'M)c(T"M)
=
(1 + d ( M ) + • • • + c n (M))(l -
C l (M)
+ • • • + (-l)"c„(M)).
It follows that (-l)kPk(M)
= (-l) 2fc c 2fc (M) + (-l) 2A =- 1 c 1 (M)c 2A _ 1 (M) + • • • +
In particular, if a sphere 5 structure, then 0 = Pm{Sim)
4m
c2k(M).
of dimension 4m admits an almost complex = 2c2m{Sim)
= 2e(5 4 m ) = 4,
which is a contradiction. This shows that a sphere of dimension 4m admits no almost complex structures. Actually, S2 and S 6 are the only spheres that admit almost complex structures; the proof involves more sophisticated topological arguments (see Kobayashi-Nomizu [9] for references). An almost complex structure on S 6 can be constructed by means of Cayley numbers, (S 6 is identified with the space of unit purely imaginary Cayley numbers), and an almost complex structure on S6 induces an almost complex structure on every hypersurface in the 7-dimensional Euclidean space via the Gauss map, see for example [9]. The compact homogeneous almost complex manifolds of positive Euler number have been classified (Hermann [6]). S 6 is the simplest such example. For 4-manifolds the existence problem has been completely solved. Namely, an oriented compact 4-manifold M admits an almost complex structure if and only if there is an element c £ H2(M; Z) such that (1.2)
c2 = 2e(M) + 3a(M),
c = w2(M)
(mod 2),
and c is given by c = Ci(M). Moreover, c = Ci(M) determines the almost complex structure uniquely up to a homotopy. For example, a product M of a compact orientable 3-manifold N with a circle S1 has e(M) = <J(M) = 0, and there is an almost complex structure such that c\{M) = 0; in this case, it is, of course, easy to construct an almost complex structure since the tangent bundle of N splits into a 2-plane bundle and a line bundle.
149
Other examples of manifolds which admit almost complex structures include even-dimensional parallelisable manifolds, symplectic manifolds, products of two contact manifolds. The existence of almost complex structures is a topological question; it is a problem of reducing the structure group GL(2n; R) of the frame bundle of a 2n-manifold to GL(n;C). However, the existence of complex structures is a much deeper question. It is a difficult problem to decide whehter a given (non-complex) almost complex manifold admits a complex structure or not. It is still an open question whether S 6 admits a complex structure or not. We recall that given two almost complex manifolds (M, J) and (M,J), a map / : M —> M is said to be holomorphic if it commutes with J and J, i.e., (1.3)
Jo/. =/.oJ.
In fact, this is the Cauchy-Riemann equations when J and J are complex structures. A holomorphic function is a holomorphic mapping into C. As I said earlier, for a generic almost complex structure on M, even locally there are no nonconstant holomorphic functions. Thus there is no complex function theory on almost complex manifolds. However, there is an abundant supply of holomorphic mappings from a disk of C into any almost complex manifold M. Let D be a unit disk in C. A holomorphic map / : D —> M is called a holomorphic disk. An important theorem of Nijenhuis-Woolf [10] says that given a point p G M and a tangent vector u G T'pM there is a holomorphic disk f:D->M tangent to u. Once we have the concept of holomorphic disk and the theorem of Nijenhuis-Woolf, we can immediately define the intrinsic pseduo-distance d,M and hyperbolicity for an almost complex manifold M exactly in the same way as in the complex manifold case, [7]. In Section 5, we discuss the hyperbolicity question using differential geometric means.
2
Canonical connections for almost Hermitian manifolds
An almost Hermitian manifold is an almost complex manifold (M, J) with a Hermitian metric g. By definition, g is a a Riemannian metric satisfying the condition (2.1)
g(Ju, Jv) = g(u, v),
u,v € TXM,
x G M.
150
Given an almost Hermitian manifold (M, J,g), it is natural to consider, among all affine connections on M, those which makes both J and g parallel, i.e., (2.2)
Vg -0,
V J = 0.
There are continuously many connections V satisfying condition (2.2). In order to make V unique, additional conditions are necessary. Let e\, • • •, e„ € T'M be a local unitary frame field. Let 81, • • •, 9n be the dual local unitary co-frame field; they are locally defined (l,0)-forms and g
= e1^ + ••• +
enen.
Then the connection form for a connection satisfying (2.2) is a locally defined 1-form w = (wj) with values in the Lie algebra u(n) of U(n); it is skew-Hermitian:
(2.3)
wj.+w?=0,
which is equivalent to (2.2). The first structure equation of the connection is given by (2.4)
d0i =
-^2u)ABj+ei,
and the second structure equation by (2.5)
d w ^ - ^ w j A w f + nj,
where © = (0 l ) and fi = (flj) are called the torsion form and curvature form, respectively. Since 0 and fi are 2-forms, they decompose into components of degree (2,0), (1,1) and (0,2).
(2.6)
0 = 0(2-°> + e*1-1* + 0 (o>2) , n = n (2 - 0) + n*1-1) + ft(0-2).
Among the connections satisfying (2.2), the most natural is the one defined by the following theorem of Ehresmann-Libermann [4] (for the proof, see [8]). (2.7) Theorem. Every almost Hermitian manifold (M, J, g) admits a unique affine connection such that J and g are parallel and the torsion 0 has no (1,1)-component. We call the unique connection defined by (2.7) the canonical connection of the almost Hermitian manifold (M,J,g). This connection introduced by Ehresmann and Libermann generalizes the connection considered by Schouten-Dantzig [11] and Chern [1] in the Hermitian case. A systematic
151
comparison of various connections on almost complex manifolds is given in Yano [12]. At this point we want to digress a little to explain vanishing of various components of the torsion form. To the almost Hermitian structure (J, g) we associate the fundamental 2-form (2.8)
$ = ^ ^ 0 ' A 0
J
,
or equivalently $(u,v) — g(u, Jv),
u,vETpM.
When this form $ is closed, (M, J, g) is called an almost Kahler manifold. If, furthermore, J is integrable, then (M, J, g) is a Kahler manifold. Then all these conditions may be tabulated as follows.
d$ = 0
almost Hermitian 0(1,1) = o almost Kahler 0 (1,1) =
0(2,0)
=
integrable J Hermitian 0(1,1)
0
0(0,2)
=
=
0
Kahler 9 = 0
As in (2.5), let il = (fij) be the curvature form of the canonical connection w. In the Hermitian case, the (2,0)- and (0,2)-components of 0, vanish. But in the almost Hermitian case, they may be nontrivial. However, in order to define the holomorphic sectional curvature of an almost Hermitian manifold, we use only the (l,l)-component ft*1-1) = ((nj) ( 1 , 1 ) ) of the curvature. Write (2.9)
= Y,R*iki9k
(n))^
AS
'-
Since we are using unitary frames, we have R
= 2 ^ 9hjRikl
fjkl
=
R
\kV
For a unit vector u = J2 u%tu w e define the holomorphic sectional curvature in the direction spanned by u to be (2.10)
3.
H(u) =
1,1
fl(n<
>(u,«)u,ti) = Y,
Rfjk^jukul.
Almost complex submanifolds
Let M' be an almost complex submanifold of (M,J); at each point pe M' the tangent space TPM' is stable under J. A Hermitian metric g on
152
M induces a Hermitian metric on M'. We shall compare the holomorphic sectional curvature of M' with that of M. Let 2n = d i m M and 2m = dimM'. We choose a local co-frame field 61, • • •, 6n in such a way that 9m+1 = 0, • • •,6 n = 0 on M', (i.e., ei, • • •, e m are tangent to M1). Let u> = (ujj) be the canonical connection on M. We restrict the first structure equation of the canonical connection of M to M'. Since 6r = 0 for r = m + 1, • • •, n, we have d f l ^ - ^ T t d ^ A ^ + e0,
(3.1)
a = l,---,m,
6=1
m
(3.2)
O = d0 r = - ^ c d £ A 0 6 + e r ,
r = m + l,---,n
6=1
While the first equation (3.1) says that u/ = (cd£)ai6=i,...iTO defines the canonical connection on M'. Since 0 has no (1, l)-components, (3.2) implies that, restricted to M', the forms w£, 1 < 6 < m
(3.3)
W = W-
u
b**ra,
Y,
(a,b = l,---,m)-
Since ujra are of degree (1,0) on M', write ro
^=x;^c-
(3.4)
c=l
As in (2.9), set Prom (3.3) and (3.4) we obtain n
(3-5)
Kid = Kid-
E
KM+
r=M+l
So, for any unit tangent vector u = YlT=i u
(3.6)
H'(u) = H(u) -
£ r=m+\
n
IE a,c=l
ft
aC«a«c|2
< H(u).
153
4.
Conformal changes
Now we consider conformal changes of an almost Hermitian manifold (M, J, g) with local unitary co-frame field 91, • • •, 6n and the canonical connection u = (w*). Let / be a real valued function on M, and consider a Hermitian metric h defined by h = e2fg.
(4.1)
Then with respect to a local unitary co-frame field ip1=efel,---,ipn
= ef9n
for h, the canonical connection %/J — (ipj) for (M,J,h) terms of w and / as follows:
(4.2)
^=u)
can be expressed in
+ {d'f-d"f)8),
so that
(4.3)
V = -^VJA<^'+$\
where $* = e ' 0 * + 2 d 7 A p \ A simple calculation gives its curvature \£: (4.4)
* = fi + d{d'f - d"f)I = n -
2(d'd"f)I
and 9&M = H ^ 1 ' -
(4.5)
2(d'd"f)I.
This is significantly simpler than conformal changes of metrics in Riemannian geometry. If v is a unit vector for the metric h = e2fg, then u = e^v is a unit vector for g. Let Hg and Hh be the holomorphic sectional curvatures for g and h, respectively, i.e., Hg(u)=g(rt1<1Hu,u)u,u),
Hh(v)=h(¥1>1Hv,v)v,v).
Then from (4.5) we have Hh{v)
= =
h{rt1'1\v,v)v,v)-2h((d'd"f){v,v)v,v) e- 0(n< >(u,«)u,ti) 2e-2f(d'd"f)(u,u). 2/
lll
Hence, (4.6)
Hh(v) = e- 2 / H f f (u) -
2e-2f{d'd"f)(u,u).
154 5.
Differential geometric criterion for hyperbolicity
One of the sufficient conditions for a complex manifold to be (complete) hyperbolic (in the sense that its intrinsic pseudo-distance is a (complete) distance) is that it admits a (complete) Hermitian metric with holomorphic sectional curvature bounded above by a negative constant. We will show that this can be generalized to the almost complex case. Essential points of the proof are Ahlfors' Schwarz lemma and the fact (shown in (3.6)) that the holomorphic sectional curvature of an almost complex submanifold of M does not exceed that of M. We recall Ahlfors' Schwarz lemma. (5.1) Theorem Let D be the unit disk {\z\ < 1}, and go the Poincare metric, normalized so that its curvature is — 1. Let g be any Hermitian pseudo-metric on D whose curvature is bounded above by —1. Then 9
<9D-
By a pseudo-metric we mean that g may degenerate at some points. At such points, the curvature is not defined. Our assumption is that the curvature is bounded by —1 only at the points where the pseudo-metric is non-degenerate. Prom (3.6) and (5.1) we obtain (5.2) Theorem. Let (M,J,g) be an almost complex manifold such that its holomorphic sectional curvature is bounded above by —1. Then every holomorphic map f:D —> M is distance-decreasing, i.e., f*9 < 9DProof. Set g = f*(g). Since g vanishes where / is degenerate, it suffice to consider the points where / is non-degenerate. Let a 6 D be such a point and U a small neighborhood so that / : U -> M is an imbedding. By (3.6) the curvature of g in U is bounded above by —1. By Theorem (5.1), 9 < 9D on U. D As a consequence we obtain the following differential geometric criterion for hyperbolicity. (5.3) Theorem. Let (M, J,g) be an almost Hermitian manifold. If its holomorphic sectional curvature is bounded above by a negative constant, then (M,J) hyperbolic. If g is moreover complete, (M,J) is complete hyperbolic Proof. By multiplying g with a suitable constant, we may assume that the holomorphic sectional curvature is bounded above by —1. Let p denote the Poincare distance on the unit disk D defined by go- Let Sg be the distance function on M defined by the metric g. Finally, let ^ M be the intrinsice pseudo-distance of (M, J)
155
Let f:D -» M be a holomorphic mapping, and g = f*g. 9 < 9D, which means that the mapping
By (5.2),
f:(D,p)^(M,SB) is distance-decreasing. We recall the following characteristic property of d,M- if 8 is any pseudodistance on M such that every holomorphic mapping f:(D,p) —¥ (M,5) is distance-decreasing, then 6 < dm- By this property of d^, we have 8g < dM-
Since 5g is a distance, dM is also a distance. If Sg is Cauchy-complete, so is dM• Given a point p 0 of an almost Hermitian manifold (M,J,g), there is a positive function / denned in a neighborhood V of p 0 such that the Hermitian metric h = e2fh has holomorphic sectional curvature < — 1. We indicate an explicit construction of such a function / . Let 9 = (01, • • •, 6n) be an adapted unitary co-frame field around p 0 . (By " adapted", we mean that the canonical connection form w = (a;!-) vanishes at po- For the existence of such frame field, see [8]). Take a complex local coordinate system z1, • • •, zn with origin at po such that 8l = dz'
at
po-
The local complex structure given by z1 • • •, zn has nothing to do with the given almost complex structure J (except at the origin po). Let r be a small positive number such that the coordinate system z1, • • •, zn is valid in the ball
c/r = {iNi2 = Ei^i 2 < r 2 }Now, we choose as / the following function defined on Ur:
Then (4.6) and calculation show that if r is sufficiently small, the holomorphic sectional curvature Hh(v) of h at po is negative, (see [8] for details). By continuity, in a small neighborhood V of po, the holomorphic sectional curvature of h remains negatively bounded. In [8] we used a slightly different function / . By (5.3) V is hyperbolic. Thus, (5.4) Theorem. Let (M, J ) be an almost complex manifold. Then every point po £ M has a hyperbolic neighborhood. In [8] I claimed that the neighborhood I constructed is also complete without valid proof. Although I believe that it is complete, the question is
156
still open. In dimension 4, Debalme and Ivashkovich [2] have established local complete hyperbolicity by a different method.
References [I] S-S. Chern, Characterisitc classes of Hermitian manifolds, Ann. of Math. 47 (1946), 85-121. [2] R. Debalme and S. Ivashkovich, Complete hyperbolic neighborhoods in almost complex surfaces, International J. Math., 12 (2001), 211-221. [3] C. Ehresmann, Sur la theorie des espaces fibres, Colloq. Intern'l C.N.R.S. Toplogie algebrique, Paris (1947), 3-35. [4] C. Ehresmann and P. Libermann, Sur les structures presque hermitiennes isotropes, C. R. Acad. Sci. Paris 232 (1951), 1281-1283. [5] M. Gromov, Pseudo-holomorphic curves in symplectic manifolds, Invent, math. 82 (1985), 307-347. [6] R. Hermann, Compact homogeneous almost complex spaces of positive characteristic, Trans. Amer. Math. Soc. 83 (1956), 471-481. [7] S. Kobayashi, Hyperbolic Complex Spaces, Springer-Verlag, 1998. [8] S. Kobayashi, Almost complex manifolds and hyperbolicity, Result. Math. 40 (2001), 246-256. [9] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, vol. 2, Wiley, 1969. [10] A. Nijenhuis and W. B. Woolf, Three integration problems in almost complex manifolds, Ann. of Math. [II] J. A. Schouten and D. van Danzig, Uber unitare Geometrie, Math. Ann. 103 (1930), 319-346. [12] K. Yano, Differential Geometry on Complex and Almost Complex Spaces, Pergamon Press, 1965.
T h o m Isomorphism of Equivariant Cohomology MEI Xiang Ming Department of Mathematics, Capital Normal University, Beijing 100037, P. R. China Abstract In this paper, we generalizes the Thom isomorphism from ordinary cohomology to equivariant cohomology.
1. Introduction Let M be a compact oriented differential manifold, iV be its closed submanifold, dim M - dim N — r, we have the cohomology exact sequences (see [1], §2) H*-1{M\N)
-4 H*(M,M\N)
3
A
H*(M)
and the Thom isomorphism H*~r{N)
4 H*{M,M\N)
»
ff*(r-LM,7,-LAfy\0
« #£(TxiV),
where T^N denote the normal bundle of N in M, it is diffeomorphism to the e-tubular neighborhood N(e) of N in M, HQ denotes the cohomology with compact support, j» denotes the zero extension in M. In this paper, we generalize the above result to the equivariant cohomology. Let G be a compact Lie group acts left on M, g be its Lie algebra, X £ g generates a vector field X*,N is the zero set of X*. The main results of this paper is to construct a concrete equivariant form with compact support $ e ZcGiT^N) such that: it has the similar properties as Thom form in the ordinary cohomology (see [4], p.32) and the correspondence [OJ] € H*(N) H+ [UJ A $] € H^aiT^N) gives the Thom isomorphism of the equivariant cohomology.
2. Equivariant Cohomology Let M be a compact oriented differential manifold, dim M = n, G is a Lie group acts left on M, G is also acting on C°°(M). For V € C°°(M), (g-
158
Let g be the Lie algebra of G, VX e 0 generates a C°° vector field X* on M: ( * » ( * ) = ^[4>exp(-eX)
• x)]e=0.
Let A(M) be the algebra of the differential forms of M, the exterior differential d and the contraction operator i(X*) are both antiderivations: it means that, for Va, /? G A(M), + ( - l ) d e s « a A d0,
d(aAP)=daAp
i{X*)(a A P) = i{X*)a A /J + ( - l ) d e g a a A t(Jf*)/9. Let Cx* be the Lie derivative with respect to the vector field X*, it satisfies the homotopy formula: Cx- - d • i{X*) + i(X*)d, so that Lie derivative is a derivation, i.e., for Va,/3 6
A(M),
£x> (a A p) = Cx* (a) A /3 + a A Cx> (/?)• If a e £ ( M ) , Cx'Ot = 0, a is called the equivariant differential form of M, denoted by a £ AG{M), AG{M) is a subalgebra of A(M). Introduces a operator ds on the algebra AG(M) such that: for Va € AG(M), dga = da — i(X*)a. Using the homotopy formula, we can prove that: dea g AG{M) and d^a = 0, for Va G > 4 G ( M ) , so that (^4G(M),dg) constitutes a complex, it is called the equivariant complex of M. This is the de Rham model of the equivariant cohomolgy of M. If a € AG{M) satisfy dga = 0, a is called closed equivariant form; if there exists 0 G AG{M), such that a = dBP, then a is called exact equivariant form. Defines the subalgebra of AG(M) as follows: ZG(M)
= {ae AG(M)
: dsa = 0},
BG{M) = {a G AG(M) : there exists /3 6 .4 G (M), such that a = dB/3}. It is evident that: BG{M) C ZG(M), HG(M) =
then ZG(M)/BG(M)
is called the Equivariant cohomology of M. Let a G . 4 G ( M ) , a'"' be the homogeneous part of degree n in a, we define
f a= f aW.
159
If a £ BG(M), f JM
a=
then a = dgp = dp -
i{X*)p,
f aW = f {dp- i(X*)p)W JM
JM
= f d^gln-i] = 0, JM
because M is compact. So that: if a G ZG(M) the integral fM a depends only the cohomology class [a] € HQ(M).
3. Kobayashi Lemma Let M be a compact oriented differential manifold, G is a compact group acting left on M . Introduce a Riemannian metric h on M, we can always choose that h is G-invariant. Let g be the Lie algebra of G. For VX e a, it generates a Killing vector field X* on M, it means that: For any two vector fields Y and Z on M , we have X* -h(Y,Z)
= h(Cx*Y,Z)
+
h(Y,Cx*Z).
Let N be the zero set of Killing vector field X*, N = \J Ni, where Ni i
is the connected component of N. According to the result of Kobayashi (see [3], p. 60), Ni is the closed geodesic submanifold of M with even codimension, dim Ni = 2(m — r^), (0 < r^ < m). Let T(M) be the tangent bundle of M, the G-invariant metric h of M determines uniquely the Levi-Civita connection V. Let the connection form of V be w, the corresponding curvature form is 0 = du + |[w, u]. The LeviCivita connection V has the following properties: For any two vector fields Y and Z, we have X*-h(Y,Z)
= h(Vx,Y,Z)
+
h(Y,Vx*Z).
Define an operator Ax* — Cx* — Vx* acting on the tangent bundle T(M), it is evident that: h{Ax.Y,Z)
+ h(Y,Ax.Z)
= 0,
so that Ax* is an antisymmetric operator with respect to the metric h. Along the zero set N of X*, X* = 0, so that Ax* \N = Cx* \N.- At the point x € N, Ax* (x) = Cx* (%) is a linear transformation of the tangent space TX(M) of M. Since the Levi-Civita connection is of no torsion, VX*Y - VyX* = [x%Y] = CX*Y, AX*Y = CX*Y - VX*Y =
Vy €
T(T(M)),
-VYX*.
For any vector field Y tangent to Ni, its integral curves belongs to Ni, so that X* = 0 along the integral curves of Y, it shows that: For VY 6
160
r(T(M)), AX*Y = £X*Y
= 0.
So that the action of the operator Ax*(x) on the subspace Tx(Ni) of the tangent space TX(M) is zero. Let T^-(Ni) denote the orthogonal complement of Tx(Ni) in TX(M) with respect to the metric h, dimT^iV-1-) = n — dimTx(Ni) — 2r, (0 < r» < m). Since Ax* is antisymmetric with respect to the metric h, Ax*(x) can be reduced to normal form on T(N) ( Ax.(x)
=
0 -ax
ax 0
\ '
Ax.(x)\ TJ-(Ni) \
0 -ari
ari 0 /
Let T^-Ni be the normal bundle of JV» in M, its fibre is T^{Ni), Vx € Nj. Restricts the Levi-Civita connection V of the tangent bundle T(M) to its subbundle T^iVj), we obtain the induced connection V"1, let the corresponding connection form be OJ-^, and the curvature form is fl1- — duj1- + |[a;-1-,a;-L]. Ax, is an operator acting on the normal bundle, the action of it on every fibre is the above linear transformation Ax, (x). Let Ni(e) be the e-neighborhood of JVj in M. When e is sufficient small, Ni(e) can be coincided to normal ball bundle B±(Ni) on Nt its fibre is a 2rj-dimensional ball on T^Ni with the center x and the radius e. dNi{e) is the boundary of Ni(x), it can be regarded as a normal sphere bundle S±(Ni) on Ni, its fibre is a (2r; - l)-dimensional sphere Si(e) = dBi(e) on T^Ni with center x and the radius e. Define a 1-form 7r on M\N
as follows:
it is evident that i(X*)7r = 0, so that d • i(X*)n = 0. Since dn(X*, Y) = X* • 7r(F) - y • 7T(X*) - TT([X*, F]),
we shall show that: i(X*)dn(Y) = dir(X*,Y) = 0. If F||X*, it is trivial; when YLX*, we can prove that [X*,F]±X, we have also dn(X*,Y) = 0. So that CX' = di(X*)ir + i(X*)dn = 0. It means that n £ AG{M). We can prove further that Cx+dn = 0, so we have also dir € AG(M).
161
Lemma.
lim I
n
=
1
where
Proof. See Kobayashi [4], pp.71-75. Let D be the covariant derivative determined by the Levi-Civita connection, the corresponding curvature form is ft — Dw, we have also the Bianchi identity DU = 0. Generalizes to the equivariant case, we can define the equivariant covariant derivate Dg and the equivariant curvature form fi0 to be
DB = D-i(x*),
ng = n + Ax*.
Similarly, we can prove the equivariant Bianchi identity DgQg = 0 (see [4], 2.8). Let $ € C[g] G , i.e., $ is a complex-valued G-invariant polynomial defined on g, we define
$(ftfl) = *(ft +AxOUsing equivariant Bianchi identity, we can prove that: $(fi g ) € ZQ{M) (see [2], 2.13). $(fi 8 ) is called the equivariant characteristic form of the tangent bundle T(M), [$(ftg)\ G HQ(M) is called the equivariant characteristic class of the tangent bundle. Especially, E(Clg) = ; A „ s/det(ilg) is called the Euler characteristic form of tangent bundle. Now we can rewrite the above Lemma in terms of equivariant terminology. Lemma. e
£ n ™ysi( e )dB 7 r ( i x where -E(n ) is the equivariant Euler characteristic form of the normal sphere bundle dNi(e).
4. The Thorn Isommorphism of the Normal Bundle Let TxNi denotes the normal bundle on iV, in M. B-L(Ni) is the corresponding normal ball bundle, it is coincided to the tubular e-neighborhood Ni(e) of Ni in M via exponential mapping. So we can denote Ni(e) by B±(Ni). The Lie group G keeps the point x E N fixed, since the Lie group
162
G keeps also the Riemannian metric invariant, so it keeps the geodesic passing through the point x and orthogonal to Ni invariant. It shows that: the Lie group G is the fibre-preserving transformation group of BA-{Ni), it is also the fibre-preserving transformation group of the corresponding normal sphere bundle S-^JVj) = dB^Ni) on Nt. Now we shall find the Thom form $ G ZcG(BA-{Ni)) of the normal ball bundle p : B^iNi) -> Ni, where the symbol C denotes compact support, such that the Thom isomorphism: %:HG(Ni)^HCG(B±(Ni)) is determined by the correspondence: [u] i->- [OJ A $]. The following relation of the cohomology in the introduction H—\M\N)
4 H*(M,M\N)
«
HG(TXN)
suggest us that: we must find first a form of M\N, obtain the required Thom form. So that: we put
then acts in by dQ to
$ = d9(p(r)^Ap'£^)), where r is distance from the point of T^-(N) to the origin x,p(r) is a C°° function of the fibre T^-(Ni) « R2ri of the normal bundle T^iNi) which satisfies the condition p(e) = 0, p(0) — 1. The reason to introduce the distance function p(r) is that: it makes the form $ to have compact support. Since G keeps the metric invariant, so that Cx*p{r) = 0, i.e., $ G
ACG(M).
Theorem 1. The differential form <J> is the Thom form of the equivariant cohomology, it has the following properties: (i) * G BCG(B±(Ni))
ZCG(B±(Ni)),
C
(ii) Let ix : B^r(N) -)• B±(Ni)
is the inclusion mapping, then we have
/ .BJ-(JVj) (iii) Let s : Nj ->• B±(Ni)
Proof, (i) It is trivial.
is the zero section, then we have
163
(ii)
[d(p(r)-^-)[2r'"ll|AJE(n^),
= lim/
where Bx(r) is the ball with the center x and the radius r. Since
so that
according the Lemma due to Kobayashi, we have JB±(Ni)
(iii)
s^
•*=mr)*m)=1'
=
^l"a
s*de[p(r)J^)As*p*E(nj)
= <*<»£) *E<#) = *,(£.) A Eft) = E(^).
The theorem is proved. T h e o r e m 2. Let UJ G Zo{Ni), then the correspondence [u] H-> [w A $] determines the Thom isomorphism of the equivariant cohomology
Z:
H^Ni)-+HZaiB^Ni)).
Proof. According to the Theorem 1 (ii)
/
p*uA$=
JBi-(Ni)
L A | JNi
L
i * $ ] = / u>,
JB^(Ni)
J
./JV;
since the above integral depends only the equivariant cohomology class, so we have the homomorphism of the equivariant cohomology 1: H'G{Ni) -> H*CG{B^{Ni)), Let w 6 ZcGiB^iNi)), ±
of Ni in B (Ni)
[u] H> [U A $].
dgu) — 0, and ATj(r) is the r-tubular neighborhood
« Wi(e), then in the B-1 (Ni)\Ni(r),
*(d^ A 4
we have
164
Notes that the zero section s(iVj) of BL(Ni) is the deformation retract of Bx(Ni),s • p ~ identity, so that w = p*s*uj + dr, but u has compact w support |nR±fjy.i = 0 ) t n e n w e have /
tj = lim /
d(-—Aw
r
-*0JBJ-(Ni)-Ni(T)
JB-^(Ni)
= I
B
-^(P*S*0J
JdB-^(Ni)
since fdN.,r\
Kd n
'
+ dT),
"gir
dr = 0, according to the Kobayashi Lemma, we have
JBHNi)U~ Jt*i^ ^ JSi (r)dgTt) S " ~ JN. E(Q^ ) ' then we obtain the another equivariant cohomology homomorphism p. : NZoiBHNi))
~> flc(JVf), M -> [ ^ y ] •
Now we shall that: the homomorphism T and p are mutually inverse, so that they are both isomorphisms. If a; € Za(Ni), according to the Theorem l(iii),
P. • T H = Pt[p* A $] = [
P E{n±)
\ = M.
If w € ZcG(Ni), notes that s(iVj) is the deformation retract of so that [td] = [p*s*w], then we have
„-
r T
^r
s*u
1
r
p*s*w
BJ-(Ni),
i
= [pVo.] A [d, ( / » M ^ ) ] = H A [1] = H , the Theorem is proved.
References 1. Atiyah, M. &; Bott, R., The moment map and equivariant cohomology, Topology, 23(1984), 1-28. 2. Berline, N. & Vergne, M., Zeros d'un champe de vecteur et classes characteristique d'equivariante, Duke Math. J., 50(1983), 537-549.
165
3. Berline, N., Getzer, E. & Vergne, M., Heat kernel and Dirac operators, Math. Wis., Vol. 298, Springer-Verlag, 1992. 4. Kobayashi, The transformation group in differential geometry, SpringerVerlag, 1972. 5. Bott, R. & Tu, L., Differential forms in algebra topology, SpringerVerlag, 1982.
Horizontally conformal F-harmonic maps * XIAOHUAN MO AND CHUNHONG YANG Key Laboratory of Pure and Applied Mathematics School of Mathematical Sciences Peking University, Beijing 100871, China E-mail: [email protected] and [email protected]
Abstract F-harmonic maps are maps which extremize the F-energy functional.
In this
article we study the F-harmonicity of horizontally weakly conformal maps and the conformal properties of F-harmonic maps. Key words and phrases: F-harmonic map, horizontally weakly conformal maps, F-tension field 1991 Mathematics Subject Classification: 58E20.
1
Introduction
Harmonic maps introduced by Eells and Sampson in 1964[5], are critical points of the 'Dirichlet' energy functional
\ f \\d
and in this respect can be seen as a generalisation of geodesies. Harmonic maps are solutions to an elliptic system of partial differential equations, which in general is non-linear. Harmonic maps are very important both in classical and modern differential geometry. 'This work is supported by the National Natural Science Foundation of China 10171002
166
167
A particularly interesting subclass of harmonic maps is horizontally weakly conformal harmonic maps. The interplay between the analytical condition (harmonicity) and the geometrical one (horizontal weak conformality) is a rich source of properties. Baird and Eells [2] studied some conformal properties of harmonic maps. They showed that if the dimension of target is equal two, then the fibers of horizontally weakly conformal harmonic maps are minimal submanifolds in the domain manifold. When the codomain is not of dimension two, R. Pantilie showed that the harmonicity for a horizontally conformal map is characterized by the property that the parallel displacement defined by the horizontal distribution (i.e. the distribution orthogonal to the fibres), preserves the mass of fibres[9,10]. This generalizes the well-known fact that a Riemannian submersion is harmonic if and only if the parallel displacement defined by the horizontal distribution preserves volumes. Viewing harmonic maps as maps which extremize the energy functional, their most natural generalization is the notion of F-harmonic maps, i.e. maps which extremize the F-energy functional[1]. They encompass p-harmonic maps [3,12] and exponentially harmonic maps[4]. In this paper we discuss the F-harmonicity of a horizontally weakly conformal harmonic map. In particular, we show the following Theorem 5 A horizontally conformal submersion with dilation A is an F-harmonic map if and only if the parallel displacement defined by the horizontal distribution preserves the mass of the fibres, where the fibres are given the mass density A 2 _ n .F'(§A 2 ). When F(t) = t and (2i) 2 /p, this theorem has been proved in [9,10] and [7], respectively. We also extend the results of conformal properties due to Baird and Eells, Baird and Gudmundsson, and Takeuchi.
168
2
T h e Fundamental Equation for a Horizontally Conformal M a p
Let F : [0, oo) ->• [0, oo) is a C 2 function such F' > 0 on (0, oo). For a smooth map <> / : (M, g) ->• (N, h) between Riemannian manifolds (M, g) and (N, h), we define the F-energy of <j> by ,|2-
L'ffl
dv
where \d(j>\ denotes the Hilbert-Schmidt norm of the differential d
> 4), (l + 2t) a (a > l.dimM =
2) and exp t, respectively. We shall say that <j> is an F-harmonic map if it is a critical point of the F-energy functional, which is a generalization of harmonic maps, p-harmonic maps or exponentially harmonic maps. Recall that a map 4> : M -¥ N between Riemannian manifolds is Fharmonic if and only if its F-tension field TF{<J>) := F' 0 ^ \
r+V
(F'
( ^ ) )
vanishes. The reader is referred to [1] for a detailed account of F-harmonic maps. Call a smooth map (j> : M -» N between Riemannian manifolds horizontally (weakly) conformal if for any point p € M which is not contained in the critical set C^ = {p 6 M\d(j)p = 0} of
>M= Ofor all Y € Kercty,}
of Ker d
or, locally,
<> / satisfies the following equations[13]
*
dxl dxi
v
'
for some function A : M -> [0, oo) called the dilation of
169
Proposition lLet <j> : (M, g) -> (N, h) be a horizontally conformal map with dilation X. Let rp be the tension field of <j) and r] the trace of the second fundamental form of the vertical subbundle of
(2)
where r is the tension field of (j>. In general, if / : R -> R, and h: M -* R then V ( / o h) = /'V/i
(3)
Note that 0 is a horizontally conformal map with dilation A, i.e. (jfh = ^29\n where % — (Kerdcf))1 is the horizontal distribution of <j>. Then we have \d(j>\2 = nX2
(4)
On the other hand, the F-tension field Tp of
- ' '(^)-v(r(M)) Substitute (3) and (4) into (5) and use (2) we have
T
" = -5F t(" -2)F' (? A! ) " " ^ &')}VA!"
F
' (1 A ! )' <6)
Since F' > 0 on (0, oo), (6) is equivalent to 2
TF
^FTW)+%
{n-2)F'^X
)-nX2F"(^)r,
0
(7)
170
By a straightforward computation, we obtain
vln
nfA^)-
Fif^j
vlnA
(8)
Substitute (8) into (7) we have (1). Notice that for F(t) — t we obviously recover the fundamental equation in terms of the tension field T.
3
An Extension of Baird and Eells' Result
Our fundamental equation (1) implies the following T h e o r e m 2 Let (Mm, g) and (Nn, h) be Riemannian manifolds and
Then two of the
following conditions imply the other. (a) <j> is an F-harmonic
map,
(b) (f> has minimal fibres; F^xi)
*s vertical.
This result can be viewed as an extension of Baird-Eells' observation that, if the dimension of target is equal two, then the fibers of horizontally weakly conformal harmonic maps are minimal submanifolds in the domain manifold, and when the codomain is not of dimension two, then the fibers of horizontally homothetic harmonic maps are minimal submanifolds in the domain manifold. Recall that a horizontally conformal map <j> : M ->• N is said to be horizontally homothetic if %V(1/A 2 ) = 0 on MQ. AS an immediate consequence of Theorem 2 we have Corollary 3[3,12]Le£ ( M m , g) and {Nn, h) be Riemannian and (j> : M —> N be a horizontally conformal submersion. harmonic if and only if the fibres of <j> are minimal in M.
manifolds Then <j> is n-
171
4
F-harmonicity of Horizontally Conformal Maps
Let <j> : (M, g) -> (N, h) be a Riemannian submersion. Ara showed that the fibers of (f> are minimal submanifolds if and only if <j> is F-harmonictl]. Hence we have the following: Proposition 4 A Riemannian submersion is F-harmonic if and only if the parallel displacement defined by the horizontal distribution preserves volumes. In the general case we have Theorem 5 A horizontally conformal submersion with dilation A is an F-harmonic map if and only if the parallel displacement defined by the horizontal distribution preserves the mass of the fibres, where the fibres are given the mass density A 2 _ n F'(f A2). Proof We shall denote by V the projection on the vertical subbundle. Put /* := A 2 - F ' ( | A 2 ) Then, for any basic vector field X on M we have V(Cx(jiu))
=
V*[(Xn)u)+n£xu]
=
(X(j,)V*LJ +
=
{X(J,)UJ -
=
(Xp)u - ng (X,
= =
fiV*(Cx^)
fj,g(X, r))u> 2 n
F,(
(Xfi)u) + X ~ g(X,
r
| ^ ) + V In F^lt))
TF)UJ
u
- fig(X, V In fj,)w
2 n
X - g(X,TF)iJ
Remark When F(t) = t, we reduce R. Pantilie's result[9,10]. Acknowledgements.
I would like to express my special thanks to Pro-
fessor J. C. Wood for his constant encouragement during the preparation
172
of this work. I would also like to thank Professor S. Montaldo for his very helpful suggestions.
References [1] M.Ara, Geometry of F-harmonic maps, Kodai Math. J., 22(1999), 243263. [2] P.Baird and J.Eells, A conservation low for harmonic maps, In Geometry Symposium Utrech 1980, Lecture Notes in Math. 894 (SpringerVerlag,1981) pp.1-25. [3] P.Baird and S.Gudmundsson, p-harmonic maps and minimal submanifolds, Math. Ann. 294(1992), 611-624. [4] L.Cheung and P.Leung, The second variation formula for exponentially harmonic maps, Bull. Austral. Math. Soc. 59(1999), 509-514. [5] J.Eells and J.H.Sampson, Harmonic mappings of Riemannian
mani-
folds, Amer. J. Math. 86(1964), 109-160. [6] S.Gudmundsson, On the existence of harmonic morphisms from symmetric spaces of rank one, Manuscripta Math. 93(1997), 421-433. [7] H.Jin and X.Mo, On submersive p-harmonic morphisms and their stability, accepted for publication in Contemporary Mathematics. [8] X.Mo, Horizontally conformal maps and harmonic morphisms, J. Chin. Cont. Math., 17(1996), 245-252. [9] R. Pantilie, Harmonic morphisms with one-dimensional fibres, Intern. J. Math., 10(1999), 457-501. [10] R. Pantilie, On submersive harmonic morphisms, In Harmonic morphisms, harmonic maps and related topics, (eds. C.K.Anand, P.Baird, E.Loubeau and J.C.Wood). Brest 1997, Pitman Research Notes in Mathematics, CRC Press (1999), 23-29.
173
[11] J. Sacks and K. Uhlenbeck, The existence of minimal immersions of 2-spheres, Ann. of Math., 113(1981), 1-24. [12] H. Takeuchi, Some conformal properties of p-harmonic maps and a regularity for sphere-valued p-harmonic maps, J. Math. Soc. Japan 46 (1994), 217-234. [13] J.C.Wood, The geometry of harmonic maps and morphisms, Preprint of University of Leeds, 34 (1999).
H A R M O N I C M A P S B E T W E E N C A R N O T SPACES
S E I K I NISHIKAWA Mathematical
Institute, E-mail:
Tohoku University, Sendai 980-8578, [email protected]
Japan
A fc-step Carnot space is defined to be a homogeneous Riemannian manifold of negative curvature obtained as a one-dimensional solvable extension of a certain graded nilpotent Lie group, called a fc-step Carnot group. This class of manifolds includes the rank-one symmetric spaces of non-compact type as examples. We discuss the asymptotic behavior of proper harmonic maps from a fc-step Carnot space to an (-step Carnot space, where k ^ I in general.
1. Introduction 1.1. Harmonic
Maps
Let M = (M,g) and M' = {M1 ,g') be Riemannian manifolds of dimension m and m', respectively, and u : M -> M' a C 2 map from M to M'. For the differential du : TM -> TM1 of u, the covariant derivative Vdu of du is defined by Vdu(X,Y)
= Vux~lTM'du(Y)
-
du(VlMY),
where V ™ and Vu ™ denote the Levi-Civita connection of M and the pull-back of the Levi-Civita connection of M' by u, and X,Y £ T(TM) are vector fields on M, respectively. Then the tension field T(U) of u is defined to be the divergence of du, that is, 771
T(U) = Trace,,Vdu = ^
Vdu(Ei,
E{),
«=i
where {Ei} are orthonormal frame fields on M. We call u a harmonic map if the tension filed r(u) of u vanishes identically. In terms of local coordinate systems (xl) and (a;'a) of M and M', the Riemannian metrics g and g' are given locally by 777-'
777
9 = Yldijdx^x^
g1 = Y^
«=1
a,P=l
174
g'a0dx'adx'0,
175
and u is written as u{x) = {ul{x\...,xm),...,um\x\...,xm))
=
{u«{xi)).
Then the energy density e(u) of u is defined to be the squared norm of du, that is,
e(u) = \du\2=£
£>V«*(«)»£g!,
i , j = l ct,/3=l
and for a relatively compact domain fl on M the energy .EQ (U) of u over fi is defined by EQ{U)
= s/»«<«'
= - I e{u)dng,
where \xg denotes the canonical measure on M given by the Riemannian metric g. Then the energy EQ(U) of u defines a functional EQ : C2(M, M1) -> R on the space C2(M, M') of C 2 maps from M to M', and the harmonic map equation T(U) = 0 is nothing but the Euler-Lagrange equation for the energy functional, which is expressed in local coordinate systems as
A«a+EE^»H=°.
a = l,...,m', (1.1)
» J = 1 /3,7=1
where A is the Laplace-Beltrami operator on M and T'g (u) denote the Christoffel symbols of the pull-back connection V" ™ . Since the harmonic map equation (1.1) is a system of semi-linear elliptic partial differential equations of second order, any C2 harmonic map u is indeed a smooth map. 1.2. Hadamard
Manifolds
A complete, simply connected, connected Riemannian manifold M of nonpositive curvature is called a Hadamard manifold. It is well-known by the Cartan-Hadamard theorem that M is diffeomorphic to the Euclidean space of the same dimension, and there is a unique geodesic joining any two points of M. We say that two unit speed geodesic rays 7 and a in M are asymptotic if there exists a positive constant C such that the distance d{^{t),a{t)) of 7(f) and a(t) in M satisfies d(j(t),a(t)) < C for all t > 0, and define a point at infinity of M to be an equivalence class of asymptotic geodesic rays in M.
176
Let M be a Hadamard manifold of dimension m and M(oo) denote the set of points at infinity of M. If we set M = M U M(oo), then it is known that M(oo) is homeomorphic to the (m — l)-dimensional sphere 5 m _ 1 and M to the m-dimensional closed ball Bm, relative to a natural topology, called the cone topology, of M which induces on M the original topology of M. We call M the geometric compactification or the EberleinO'Neill compactification of M and M(oo) the idea/ boundary or the sphere at infinity of M ([*]). Example 1.1. (the real hyperbolic plane) TTie most familiar example of a Hadamard manifold is the real hyperbolic plane RH2, the ball model of which is defined to be the open unit disk equipped with the Poincare metric gp of constant negative curvature —1, that is, RH2 = f{z G C | |*| < 1}, gP
4\dz\
(i-N
2N2
In this model, two geodesic rays in M = RH2 are asymptotic if and only if they meet the boundary circle S1 = {z £ C \ \z\ = 1} at the same point, and hence the ideal boundary M(oo) can be identified with Sl. Note that the Poincare metric gp blows up at the boundary circle S1 as \z\ —> 1 so that the corresponding Laplace-Beltrami operator A is degenerate everywhere on the ideal boundary Sl. The circumstance can be seen more clearly if we take the upper half plane model H+ of the real hyperbolic plane RH2, that is, 9
(
rr 2 r H +=({z
i—r
= x + V^ly
This model is isometric to RH2 RH2 given by
^i
„-,
G C | y > 0}, gH =
dx2 + dy2
^ -
under the Cayley transform $ : H^
$( Z ) = v ^ i i - X — ,
zeHl,
r2
(1.2)
which maps the real axis {z e C \ y = 0} into the boundary circle S1. Therefore we may regard $ as defining a coordinate neighborhood of RH2 around its ideal boundary S1. Hence, by taking a family of these Cayley transforms as a system of coordinate neighborhoods at the ideal boundary, the geometric compactification M = RH2 U S 1 of the real hyperbolic plane RH2 admits a structure of smooth manifold with boundary, relative to which M is diffeomorphic to {z £ C \ \z\ < 1}. Then the Poincare metric gH of the upper half plane model H2, yields a local coordinate representation of
177 the Poincare metric gp of RH2 given by $ , that is, (
with respect to a coordinate
4\dz\2
\ _
* V(l-|z|»)V-
dx2+dy2 2
y
neighborhood
'
which shows more exactly how the metric on the real hyperbolic plane
{1 d)
-
RH2
blows up near the ideal boundary as y -> 0.
1.3. Proper
Harmonic
Maps
Let M = ( M , g) and M' = (M1, g') be H a d a m a r d manifolds of dimension m and m ' , respectively, and M = M U M(oo) and W = M'U M'(oo) be their geometric compactihcations. Let u : M —> M' be a proper harmonic m a p from M t o M'. Here the properness of u means t h a t if {pj} is a sequence of points in M diverging to the ideal boundary M(oo) of M, then {u(pj)} also diverges in M' to the ideal boundary Af'(oo) of M'. Suppose now t h a t u extends to a continuous m a p u : M —> M' between the geometric compactihcations. Since u is assumed proper, it follows t h a t u maps the ideal boundary M(oo) into the ideal boundary M ' ( o o ) , and thus we have the boundary value / = it|M(oo) : M(oo) -> M'(oo) of u between the ideal boundaries. Our primary objective is to investigate to what extent the asymptotic behavior of a proper harmonic m a p u near the ideal b o u n d a r y is determined by its boundary value / . T h e first remarkable result in this regard was obtained by Li and Tarn [4] in the case where M and M' are real hyperbolic spaces RHm and RHm . Namely, identifying the ideal boundaries of RHm and RHm with Sm~l and Sm - 1 , respectively, they proved the following rigidity at infinity for proper harmonic maps with C 1 boundary value. T h e o r e m 1.1. /*/ Let u be a proper harmonic map from RHm to RHm , which extends to a C1 map between the geometric compactifications. In the ball model, in terms of the polar coordinates (p,rj) = (p, n1,. .. ,77 m _ 1 ) on Bm and (r,8) = ( r , 8 1 , . . . ,6m _ 1 ) on Bm , express u as u{p,rj) = (r(p,rj),9(p,rj)). Let f : 5 m _ 1 —> Sm ~x be the boundary value of u, and assume that the energy density e(f) of f, relative to the canonical metrics on S"1^1 and Sm ~1, is nowhere vanishing. Then, on 5 m _ 1 , u must satisfy
£ = J«L,
£-0,
op
dp
Vm- 1
.<«<.M
d.4)
178
It follows from (1.4) that the expression u(p,r]) = (r{p,rj),6(p,r])) has the following expansion near the ideal boundary 5 m _ 1 as p —> 1: r(p,r,) = 1 - (1 - p)v/e(f)(r1)/(m-l)+ f(M)=f(i))+o(l-p),
o(l - p),
l
Prom this expansion, Li and Tarn proved the uniqueness of proper harmonic maps with C1 boundary value. To be precise, let u and v be proper harmonic maps from RHm to RHm , which extend to C 1 maps between the geometric compactifications. If u and v have the same boundary value / on the ideal boundary S™ - 1 and the energy density of / : S"1"1 ->• Sm ~l is nowhere vanishing, then u and v are identical. It should be remarked, however, that for proper harmonic maps u : RHm ->• RHm' which are not C 1 up to the ideal boundary Sm~1, we do not in general have the uniqueness. Indeed, Li and Tarn [4] constructed a two-parameter family of harmonic diffeomorphisms of RH2 that are only 1/2 Holder continuous at a boundary point and smooth everywhere else, and whose boundary value is the identity map of S 1 . 2. C a r n o t Spaces 2.1. Homogeneous
Manifolds
of Negative
Curvature
In order to study the asymptotic behavior of proper harmonic maps, we now review briefly the geometry of homogeneous Hadamard manifolds. Let M = (M, g) be a Hadamard manifold of dimension m with the ideal boundary M(oo), and suppose M to be homogeneous, that is, the isometry group I{M,g) of M acts transitively on M. Then it is known that there exists a solvable subgroup G of Jo (M, g), the identity component of I(M, g), that acts simply transitively on M ([3]). Therefore we can identify M with a solvable Lie group G and the Riemannian metric g on M with a left invariant metric ( , ) on G. Consequently, any homogeneous Hadamard manifold M can be represented as a simply connected solvable Lie group G endowed with a left invariant metric ( , ) of nonpositive curvature. For instance, if M is a rank one symmetric space of noncompact type, then I0(M,g) has the Iwasawa decomposition N • A • K, where N is a nilpotent subgroup, A is an abelian subgroup and K is a maximal compact subgroup, respectively. Since K is known to be the isotropy subgroup of a point in M, by setting G — N • A, the semi-direct product of N with A, we obtain a solvable Lie group G acting simply transitively on M. If we further assume that M is of strictly negative curvature, then it follows that G is a one-dimensional solvable extension of a nilpotent Lie
179 group TV. Indeed, let g denote the Lie algebra of G and n = [Q,B] be its derived algebra. Then we know, since g is solvable, that n is a nilpotent subalgebra of g. Moreover, it follows from the curvature condition that the orthogonal complement n x of n must be one-dimensional, that is, n x = R{H} with a choice of a generator H (see [3] for details). Since G is a simply connected solvable Lie group, this implies that G is a semi-direct product TV xi i? of a nilpotent Lie group TV with the abelian group R, where TV = [G, G] is the commutator subgroup of G, which is a simply connected subgroup with Lie algebra n. As a result, G is diffeomorphic to the product manifold TV x R of TV and the real line R, and hence, identifying R with the positive half line R+ = {y € R \ y > 0} by the exponential function R3 s >-> y = es £ R+, we obtain a diffeomorphism * :TV x R+ 9 (n,y) ^-n-logy
€ G = N x R,
(2.1)
which identifies G with the half space TV x R+, the product manifold of TV and R+. Consequently, a homogeneous Hadamard manifold M of strictly negative curvature can be represented as the half space TV x R+ of a simply connected nilpotent Lie group TV with the positive half line R+, which we call a half space model of M. Moreover, we may regard the diffeomorphism $ as a generalized Cayley transform. Indeed, if M is the real hyperbolic plane RH2, then TV is isomorphic to the abelian group R, so that the product manifold TV x R+ is identified with the upper half plane model H2+ of RH2, and the diffeomorphism $ in (2.1), composed with the diffeomorphism given by the simply transitive action of G on RH2, is identical with the Cayley transform $ : H\ -> RH2 in (1.2). 2.2. Rank
One Symmetric
Spaces of Noncompact
Type
In the case of a rank one symmetric space of noncompact type, we see that the nilpotent group TV appearing in the half space model is very restricted. To be precise, let M be a rank one symmetric space of noncompact type, or equivalently a connected symmetric Riemannian manifold of strictly negative curvature, and identify M with a solvable Lie group G with left invariant metric. Recall that the curvature tensor field R of M is parallel with respect to the Levi-Civita connection V of M, that is, VR = 0. Then it follows from this curvature condition that the nilpotent group TV in the half space model TV x R+ of M must be a 2-step nilpotent group at the most. Namely, the Lie algebra n of TV satisfies
[n,[n,n]] = {0}.
180
Consequently, if we set ri2 = [n, n] and ni to be the orthogonal complement of ri2 in n, then the Lie algebra g of G is decomposed into g = n1+n2
+
R{H},
where H denotes a generator of n1-, and n — ni + rt2 becomes a graded Lie algebra. Indeed, letting it; = {0} for i > 3, we have K.rtj] C ni+j,
l
Moreover, we see that ni and n2 are in fact the eigenspaces of the adjoint representation adif of H on n with eigenvalues A and 2A for some nonzero constant A £ R, respectively: m = {X <En\a,dH{X) = i\},
i = 1,2,
and each rank one symmetric space of noncompact type is distinguished by the dimension of ri2, which is equal to 0,1,3 or 7 corresponding to real hyperbolic spaces RHm, complex hyperbolic spaces CHm, quaternion hyperbolic spaces HHm, or the Cayley hyperbolic plane C&H2, respectively. For details, we refer the reader to Heintze [3]. Example 2.1. (complex hyperbolic spaces) Note that, in this context, real hyperbolic spaces are exceptional in the sense that n is abelian, that is, 1-step nilpotent so that n = ni and ri2 = {0}. The nature of the geometry in the case where n is 2-step nilpotent becomes apparent once we look at complex hyperbolic spaces. More precisely, let M be the complex hyperbolic space CHm of dimension m, which is defined to be the open unit ball B2m in the complex Euclidean space Cm of complex dimension m equipped with the Bergman metric QB of constant holomorphic sectional curvature —1, that is,
CH~ = L e c- 11.| < 1}, „ = l ^ ' - ' ^ - ' ^ W ) . As in the case of real hyperbolic spaces, in this model the ideal boundary M(oo) of M is identified in a natural way with the boundary sphere S2m~1 of the ball B2m. Also, it is well-known that the special unitary group 5(7(1, m) of signature (l,m) acts transitively on B2m as isometries, and the isotropy subgroup K at the center of B2m is isomorphic to U(m), which is a maximal compact subgroup in the Iwasawa decomposition of 5(7(1,m). Indeed, the decomposition is given as 5(7(1, m) = N • A • K, where N is a 2-step nilpotent subgroup, called the Heisenberg group, and A is a onedimensional abelian subgroup. By setting G = N x A, we then obtain a
181
solvable subgroup G of SU(l,m), which acts simply transitively on B2m as isometries, so that CHm is identified with a solvable Lie group G with a left invariant metric ( , ) . Let N x R+ be the half space model of CHm obtained from the above identification, and $ : TV x R+ —• CHm the corresponding generalized Cayley transform. Let n = ni + n2 be the graded Lie algebra decomposition of the Lie algebra n of N. Then, under the left translations of N, ni and ri2, being identified with subspaces of the tangent space of N at the identity, define complementary distributions on N. Furthermore, under the generalized Cayley transform $, the ideal boundary M(oo) = 5 2 m _ 1 of CHm is identified with the one-point compactification of the Heisenberg group N, and if we consider the Hopf fibration S2m~l -»• CP171"1 of S2m~l over the complex projective space CPm~l of dimension m — 1, then the distribution 112 corresponds to the vertical subspaces along the fiber and ni to the distribution of horizontal subspaces defined by the canonical contact structure on the odd-dimensional sphere 5 2 m _ 1 . See j5] for more detail. 2.3. k-step
Carnot
Spaces
Motivated by these observations, we now consider a more general class of homogeneous Riemannian manifolds of negative curvature that arises as a one-dimensional solvable extension of certain k-step nilpotent Lie groups, called Carnot groups. More precisely, let G be a simply connected, connected solvable Lie group satisfying the following conditions: (1) G is a semi-direct product of a nilpotent Lie group N and the abelian group R. (2) If n and g = n + R{H} denote the Lie algebras of N and G, respectively, then n has a decomposition n = J2i=i ni m t o ^ subspaces given by ^ = {X e n I adH(X) = i\X},
i = l,...,k,
(2.2)
for some nonzero constant X £ R. Note that, since the adjoint representation adif is a derivation of n, the above decomposition of n defines a graded Lie algebra structure of n, that is, [ni,r\j]Cni+j,
l
(2.3)
with the convention n* = {0} for i > k. In particular, n is a fc-step nilpotent
182
Lie algebra, that is, the k-th derived algebra of n vanishes: [n,---[n,[n,n]]---] = {0}. Moreover, since the eigenvalues of adH are nonzero real numbers, it follows from a theorem of Heintzef3] that G admits a left invariant metric g of strictly negative curvature. Consequently, we obtain a homogeneous Riemannian manifold M = (G, g) of strictly negative curvature. With these understood, we give the following definition. Definition 2.1. A homogeneous Riemannian manifold M — (G,g) of negative curvature obtained as above is called a k-step Carnot space. For example, real hyperbolic spaces are 1-step Carnot spaces, and complex or quaternion hyperbolic spaces and the Cayley hyperbolic plane are 2-step Carnot spaces. Also, it should be noted that if k > 2, then a /c-step Carnot space is not symmetric (cf. Subsection 2.2). Remark 2.1. As our convention in the following, for a given k-step Carnot space M = {G,g), we always choose H, the generator of the Lie algebra of the abelian group R, in such a way that it is of unit length, that is, g(H,H) = 1, and with respect it the eigenvalues iX of the adjoint representation adH are all positive, that is, A > 0. Now, let M = {G,g) be a &-step Carnot space, where G is a simply connected, connected solvable Lie group which is a semi-direct product N x R of a nilpotent Lie group N with R, and g is a left invariant metric on G of strictly negative curvature. Recall that, as seen in Subsection 2.1, M is represented as the product manifold N x R+ of N and the positive half line J R + via a generalized Cayley transform $ : TV x R+ —¥ G given by $ : N x R+ 9 (n, y) i-> n • exp sH € G = N x R,
(2.4)
where s = logy and H is chosen as in Remark 2.1. This half space model N x R+ of M is convenient to study the geometry of M at infinity. In fact, since M is a Hadamard manifold, we have the ideal boundary M(oo) and the geometric compactification M — MuM(oo) of M, respectively. Then, regarding a generalized Cayley transform $ as defining a coordinate neighborhood around the ideal boundary M(oo), we obtain the following proposition which describes the behavior of g near M(oo). Proposition 2.1. f'] Under a generalized Cayley transform $ : N x R+ —> G, the metric g is expressed on a half space model N x R+ of M as a k-ply
183
warped product metric 1
1 + ^9n2+---
* 9=^9n1
1 +^ 9 n
k
dy2 + ^ ,
, s (2.5)
where y denotes the coordinate on R+! A is given in (2.2) and gni + gn2 + • • • + gnk is a left invariant metric on N. As a consequence, we see that, on the half space model N x R+ of M, the eigenvalues {i\ | i = 1 , . . . , k} of the adjoint representation adiJ in (2.2) indeed reflect on the growth order of the metric g as y —> 0. Example 2.2. (1) In the case of the real hyperbolic space RHm, as seen in Subsection 2.2, N is abelian so that k = 1, and with respect to the canonical choice of H, we see that A = 1 and the Poincare metric gp is expressed as
$*gp = ±[(dxl)2 + --. + (dxm-iy] + d^, y y where (x1,... ,xm~l ,y) is the canonical coordinate system of the upper half space model Rm~l x JR+ =* N x R+ of RHm (cf. (1.3);. (2) In the case of the complex hyperbolic space CHm, as seen in Example 2.1, N is the Heisenberg group so that k = 2, and with respect to the canonical choice of H, we see that A = 1/2 and the Bergman metric gs is expressed as .. m~ 1 y
j=l
.. y
m—1 1=1
, 2 y
where (z1,..., zm~l, t) is the canonical coordinate system of the Heisenberg group N £ Cm~l x R, and dzl and dt + ^,7=^ V^l{zidzi - zidzi)/2 give rise to left invariant 1-forms on N if"]). It is immediate from (2.5) that on a half space model N x R+ of M, the unit vector field X = y(d/dy) along R+ directions {(n,y) \ n = const} satisfies VxX = 0 and hence defines geodesic rays which are asymptotic each other. Also, since M is of strictly negative curvature, we see that any two distinct points in M(oo) can be joined by a unique geodesic of M Q1]). As a consequence, the ideal boundary M(oo) of M is identified with the one-point compactification of the Carnot group N. Namely, we have the following Proposition 2.2. On a half space model N x R+ of a k-step Carnot space M, R+ directions define asymptotic geodesic rays in M and hence gives rise to a point at infinity oo € M(oo). Moreover, M(oo) \ {oo} is naturally identified with N x {0}.
184
2.4. Smooth Structures
on
M(oo)
It should be remarked that in the definition of a generalized Cayley transform $ : N x R+ ->• G given in (2.1) and (2.4), we identify the real line R with the positive half line R+ by the exponential function R 3 s ^ y = esG R+. However, for a given fc-step Carnot space M = (G,g), the choice of a diffeomorphism between R and R+ is not unique, and in general it is not apparent a priori which choice should be canonical or expedient. In fact, for a nonzero constant a 6 it, if we choose the exponential function R 3 s H-> y — eas € -R+ in (2.4), then the pullback metric <&*g in Proposition 2.1 takes the form 1 **9=^9n1
1 + ^gn2
1 1 dy2 + --- + ^j^;gnk + ^ ^ ,
(2.6)
and hence the metric g blows up at the ideal boundary M(oo) of M with different growth orders from those in (2.5). Moreover, when taking the corresponding generalized Cayley transforms $ : N x R+ 4 G as local coordinate neighborhoods around the ideal boundary M(oo), these different choices of exponential functions identifying R with R+ define different smooth structures on N x R+ which are not diffeomorphic at the ideal boundary N x {0}, since the coordinate change from y = es to y = eas is not diffeomorphic at y = 0 unless a — 1. Remark 2.2. For a given nonzero constant a £ R, if we take A/a in place of X in (2.2), then it is evident from Proposition 2.1 that on the half space model N x R+ defined by the generalized Cayley transform (2.4), we obtain the pull-back metric $*g that blows up at the ideal boundary M(oo) with the same growth orders as those for the metric in (2.6). Since different choices of the eigenvalues of the adjoint representation adH imposed in the condition (2.2) of the Carnot group G give rise to different smooth structures at the ideal boundary N x {0} on the half space model N x R+ of the Carnot space M = {G,g), it also affects considerably the asymptotic behavior of proper harmonic maps between Carnot spaces. Note here that these different smooth structures are not diffeomorphic each other only on the boundary N x {0}. Before going into details in the next section, we first remark the following typical phenomenon. We now take M to be the complex hyperbolic space CHm of dimension m and M' to be the real hyperbolic space RHm of dimension m', and let M = {G,g) and M' - {G',g') be the representations of M and M' as Carnot spaces, respectively. Recall that the Lie algebras g and g' of G and
185
G' are decomposed respectively as g = ni+n2
+ R{H},
g' = n[ +
R{H'},
and suppose on their half space models N x R+ and N' x R+ that the pull-back metrics $*g and $'*g' are expressed as 1
1
**9 = ^
A
+^
2
d
1
m
"^
A
a
'2
+ f, *V = ± £ (
according to the choices of generalized Cayley transforms $ : N x R+ -> G and $ ' : AT' x U + -» G', which are not necessary to be canonical (cf. Proposition 2.1 and Example 2.2). When we take the generalized Cayley transforms $ and $' in the canonical fashion as in Example 2.2, that is, in the case when A = 1/2 and /i = 1, the following non-existence theorem has been proved by Ueno [7]. Theorem 2.1. f] Let A = 1/2,/x = 1 and m,m' > 2. Then there exists no proper harmonic map u : M —> M1 which extends to a C1 map up to the ideal boundary. However, if we choose A = /x = 1 and take the corresponding generalized Cayley transforms $ and $', then $ induces a new smooth structure on the half space model A^ x R+ of M, which is not diffeomorphic to the canonical one on N x {0}, and we have many proper harmonic maps from M to M'. Namely, the following existence theorem was proved by Donnelly [2]. Theorem 2.2. f2] Let A = /i = 1. Then, for any C1 map / between the ideal boundaries M(oo) = 5 2 m _ 1 and M'(oo) = S™''1 with nowhere vanishing energy density e(f), there exists a proper harmonic map u : M —> M' which assumes f continuously. 3. Asymptotic Behavior of Proper Harmonic Maps 3.1. Adapted
Frame
Fields
Let M = (G, g) be a fc-step Carnot space of dimension m and M' = (G' ,g') an Z-step Carnot space of dimension m', respectively. Recall that G and G' are simply connected, connected solvable Lie group that are semi-direct products G = N x R and G' = N' x R, where A' is a fc-step Carnot group and N' is an /-step Carnot group, and g and g' are left invariant metrics of strictly negative curvature, respectively. Also, by definition, the Lie algebras n and n' of N and N' are assumed to have graded Lie algebra
186
decompositions k
n = Y^ rii,
rij = {X 6 n | adiJ(X) = i\X},
i = 1 , . . . , k,
(3-D
T n
n
n
' = X^ i '
'i = (
I e n
' l adff'(X) = i/iX},
j = l,..., I,
3= 1
where the generators H and H' of the Lie algebra of the abelian group R are chosen as in Remark 2.1. Correspondingly, under generalized Cayley transforms <E> : N x .R+ -> G and $ ' : TV' x R+ ->• G', the pull-back metrics $*<7 and $'*
1 +••• + ^ 9 n
k
dy2 + ~r, (3.2)
5
y/2/i ffn; +
yl4lJ,
9n'2-\
I" ,2,,J 9n[ +
where y and y' denote the coordinates on their respective positive half lines R+, respectively. As in the case of complex hyperbolic spaces we looked at in Example 2.1, under the left translations of N, each subspace n* in the graded Lie algebra decomposition of n in (3.1) defines a distribution on N, that is, a subbundle of the tangent bundle of N. Indeed, each n;, being identified with a subspace of the tangent space of N at the identity, defines by left translations of N a subspace (rii)p of the tangent space of N at each p € TV. We denote these distributions on N also by n*, 1 < i < k, which may be regarded, under the identification of N x {0} with M(oo) \ {oo} as in Proposition 2.2, as defining a geometric structure on the ideal boundary Af(oo) of M. Similarly, we obtain the distributions nj, 1 < j < I, on N', which also define a geometric structure on the ideal boundary M'(oo) of M'. Now, let u : M -» M' be a proper C°° map from M to M'. Suppose that u extends to a continuous map u : M —> M' from the geometric compactification M = M U M(oo) of M to the geometric compactification M' = M' U M'(oo) of M'. We are then interested in necessary conditions to be satisfied by the boundary value / = u|Af (oo) : M(oo) -)• M'(oo) of u at the ideal boundary M(oo), when u is a harmonic map in the interior M.
187
In order to carry out computations in terms of local expressions, we take the half space models N x R+ and N' x R+ of M and M', and identify, via the corresponding generalized Cayley transforms, the geometric compactifications M and M 7 with (N x [0, oo)) U {00} and (N' x [0,00)) U {00'}, respectively. Here 00 G M(oo) and 00' E M'(co) denote the points at infinity defined by R+ directions in N x R+ and N' x R+ (see Proposition 2.2), respectively. We then define adapted frame fields {e^} on N x R+ and {e'a} on N' x R+ in the following way: Let eo = d/dy and ej, = d/dy'. Set n^ = dimn^ and n'P = dimnp, where 1 < A < A; and 1 < P < I. For each r u , choose an orthonormal basis { e ^ } , ! < i < UA, relative to the left invariant metric gnA, and extend them to left invariant vector fields on N. Note that, since we have [ n ^ n g ] C XIA+B a s in (2.3), we may write the structure constants of N as nA+B
[eAi,eBi]=
^2 aAA^)re{A+B)r,
l
and all other bracket products are trivial. Similarly, for each n'P, choose an orthonormal basis {e'p }, 1 < a < n'p, relative to the left invariant metric g'n, , and extend them to left invariant vector fields on N', with respect to which the structure constants of N' may be written as n
P +Q
7=1
all other bracket products being trivial. With respect to these frame fields, we denote the differential du and the tension field T(U) of u as
du
m —1 m'— 1
e
r u
m' — 1
= E E < * ® «> ( ) = E
ei
a=0
respectively, where {e,*} denotes the dual coframe field of {ej}. Also, as usual we denote the components of the metric g by gij — g(ei,ej) and the inverse matrix of (gij) by (g*-7). In terms of these adapted frame fields, as a direct consequence of (3.2), we have the following local expression for the tension field T(U), which will be convenient to investigate the asymptotic behavior of proper harmonic maps between Carnot spaces. Proposition 3.1. Let u : M —> M' be a C2 map from a k-step Carnot space M to an l-step Carnot space M1, which maps N x R+ to N' x R+.
188 Then, with respect to adapted frame fields, the local expression r(u)a tension field r(u) of u is given by
of the
m —1
A
i«)° = E s"(ei •«?) + (i - A E
• n*)y<
=0 771 — 1
u
(y'ou)-^9 (u°)2 m —1 i=0
I
np
P-i
/3=1
(3.3)
k
771 — 1
9U(ei • «f°) + (l - A E ^ ' ^ ) r f
T(U)P° = E ?=0
,4-1 771 — 1
2=0 n
-P
Q
n
P-f-Q k(*'+Q)-r„,Q/3„.( p +Q)-,
j=0
Q=l
where 1 < P < I and I < a
0=1
7=1
It should be remarked that in (3.3) the fourth term of T(u)Pa does not appear when P = I. 3.2. The case of X = fi To clear up the nature of the problem, let us first look at the case where A = /i. Namely, we suppose that on the half space models N x R+ and N' x R+ the pull-back metrics $*g and $'*g' in (3.2) blow up with the same growth orders at the boundaries N x {0} and N' x {0}, respectively. Also, for the sake of simplicity of argument, we set A = \i = 1. Let u : M —> M' be a proper C°° map from a fc-step Carnot space M to an /-step Carnot space M'. Suppose that u maps N x R+ to N' x R+ and extends to be a C 1 map up to the boundaries TV" x {0} and N' x {0}, that is, we now suppose that ueC°°(N
xR+,N'
x J ? + ) n C 1 ( 7 V x [0,oo),N' x [0,oo)).
Then we investigate to what extent the growth orders of the metrics $*
189 First, with respect to adapted frame fields, we look at the e'0 component T°(U) of r(u). For each integer 1 < B < I, multiply the formula for T°(U) in (3.3) by (y' o u)2l~1y~2B and let y —> 0. Then we obtain Lemma 3.1. Let u : M —> M' be a proper C°° map from a k-step Carnot space M to an I-step Carnot space M', which maps N x R+ to N' x R+. Suppose that u extends to a Cx map up to the boundaries N X {0} and iV'x{0}. Then, for each 1 < B < l, with respect to adapted frame fields, the local expression r(u)° of the tension field T(U) of u satisfies the following. (1) As y —>• 0, the first three terms of T(U)° X (y1 ou)2l~ly~2B converges to 0 if B < I, and to
~{J2AnA)(u°0)21
ifB = l.
A=l
(2) The fourth term of T(U)° X (y' ou)2l~1y~2B
P=l-B+1
/3=1 (
m(B,P)
+ E P=l-B+1
where m(B,P)
is given by
nA
rip
E ^£E(^-V°«) , - p «2) 2 +°(i). A=l
i = l /3=1
= min{B + P -
l,k}.
Since r(u)° = 0 if u is a harmonic map, from Lemma 3.1 with B — 1, we can easily deduce the following condition satisfied by the Cl boundary value of a proper harmonic map. Lemma 3.2. Under the assumption of Lemma 3.1, ifu is a harmonic map, then it satisfies the following conditions on the boundary N x {0}. (1) If 1 = 1, then
( E ^ ) K ) 2 = EEK lp ) 2 A=l
i=0 0=1
(2) If I > 1, then for each 1 < i < n\ and 1 < /3 < n\ ulg = 0,
uf. = 0.
For instance, if u is a harmonic self-map of the complex hyperbolic space CHm, then, since k = I = 2 and n'2 = 1, Lemma 3.2 (2) states that u2 = 0
190 for 1 < i < n\ provided y = 0. Namely, in terms of the half space model N x R+ of CHm, the C1 boundary value / of u satisfies dfP({ni)P) C ( n i ) / ( p ) j 2 1 1
p£Nx{0},
2m_1
which means that / : S " " -+ 5 is a contact transformation on the ideal boundary 5 2 m _ 1 (cf. Example 2.1 and [2]). It should be remarked that in general Lemma 3.2 (2) implies Corollary 3.1. Let I > 2 and u : M -+ M' be a proper harmonic map from a k-step Carnot space M to an I-step Carnot space M', which maps N x R+ to N' x R+. Suppose that u extends to a Cl map up to the boundaries N x {0} and N1 x {0}. Then the boundary value f of u satisfies i-i
dfP((m)p) C XlK'W) for any p e J V x {0}. If a given proper harmonic map u has higher differentiability up to the boundaries, then we have further control of the asymptotic behavior of u at the boundary. More precisely, we obtain the following necessary conditions involving higher order derivatives of u at the boundary, which can be deduced from Lemma 3.1 with an induction argument. Proposition 3.2. Let I > 2 and 1 < r < I - 1. Let u : M -> M1 be a proper harmonic map from a k-step Carnot space M to an I-step Carnot space M', which maps N x R+ to N1 x i ? + . Suppose that u extends to a Cr map up to the boundaries N x {0} and N' x {0}. Then the following hold on N x {0}. (1) For each l-r + l
e0c.((y'o«)'-p^)=0,
0
(2) For each I - r + 1 < P
e?-((z/'°w/~ P u2) = 0 ,
0
Corollary 3.2. Under the assumption of Proposition 3.2, if in particular UQ 7^ 0 on the boundary N x {0} ; then the following hold on N x {0}. (1) For each l-r + l
0<s
+ l
P
e 0 • u /. = 0,
+ r-l. min{P -l + r, k},
0 < s < P + r - A - I.
191
Regarding the derivative UQ of u at the boundary, we can also deduce from Lemma 3.1 the following Lemma 3.3. Let u : M —> M' be a proper harmonic map from a k-step Carnot space M to an I-step Carnot space M', which maps N x R+ to N' x R+. If u extends to a Cl map up to the boundaries N x {0} and N1 x {0}, then it satisfies on N x {0}
( £ AnA) K)2/ - J2 p i > -1)!}-2 (4-1 • ((,' o uy-pu?))2 A=\
P=l I
min{P,A;}
0=1
nA
n'P
i=l
0=1
- E E pEY:{(l-Ayr2{ei0-A-((y'oUy-pu2)) P=\
A=l
=o.
In the case where k > I > 2, we can easily see from Lemma 3.3 that if
EE( i=l
u
'f)
2
^
0
on7Vx{0},
(3.4)
0=1
or equivalently the boundary value / of u satisfies dfP((ni)P)tJ2(n'i)f(p)>
P£Nx{0},
(3.5)
then UQ 7^ 0 at the boundary JV x {0}, since min{P, k} = P. Combining this with Corollary 3.2 with r = I — 1, we then obtain the following result. Proposition 3.3. Let k > I > 2. Let u : M —> M' be a proper harmonic map from a k-step Carnot space M to an I-step Carnot space M', which maps N x R+ to N' x R+. Suppose that u extends to a Cl map up to the boundaries N x {0} and N' x {0}, and satisfies (3.4) or equivalently (3.5). Then the following hold on N x {0}. (1) For each
2
UQ0
= 0,
(2) For each 2
0 < s < P - 2, 1 < 0 < n'P.
and 1 < A < min{P - 1, k}, 0<s
Now, we turn to other components T{U)PCX of the tension field T(U) to obtain further necessary conditions to be satisfied by the boundary value of a proper harmonic map u : M -> M'. To this end, in the remainder of
192
the argument, we suppose that u extends to a Cl map up to the boundaries N x {0} and N' x {0}, and satisfies
t6-0)
i=l 0=1
u°0^0
ifk
on N x. {0}. Namely, it is assumed in both cases that ujj ^ 0 on the boundary N x {0}. Under these assumptions, we multiply the formula for T(U)P" in (3.3) by (y' o u)2l~p~ly~2l+1, and let y —> 0. Then, invoking Corollary 3.2 and Proposition 3.3, we can deduce Lemma 3.4. Let u : M —> M' be a proper C°° map from a k-step Carnot space M to an l-step Carnot space M', which maps N x R+ to N' x R+. Suppose that u extends to a Cl map up to the boundaries N x {0} and N1 x {0}, and satisfies (3.6) on N x {0}. Then, as y —> 0, the local expression T(U) a of the tension field T(U) of u satisfies the following. (1) For 1 < P < I, the first three terms ofT(u)Pa x (y' ou)2l~p~ly~2l+l converges to
~{(P ~ I)!}"1 ( E AnA + p) (4)2l-P~1
(e^1 • uP«).
A=l
(2) For 1 < P < I - 1, the fourth term of converges to
T{U)P«
X
(y1
oU)2l'p~1y-2l+1
Q=l
0=1 7=1 l-P
m\n{Q,k)
+E
ci{P,Q,A)(ul)2l-p-2Q-1
E
Q=l HA
A=l n
Q
n
P+Q
x E E E i>Z£h^~A •«??)(ep+Q~A • « r o ) l ) . i=l /3=1
where Cl(P,Q,A)
7=1
= {(Q-A)l(P
+
Q-Ay.}-1.
193
As an immediate consequence of this Lemma, we then obtain the following conditions satisfied by the Cl boundary value of a proper harmonic map, since T(u)Pa = 0 if u is harmonic. Corollary 3.3. Under the assumption of Lemma 3.4, if u is a harmonic map, then it satisfies on the boundary N x {0}, 4 " 1 • u{,° = 0,
l
and
{(p -1)!}- 1 ( E
An
* + p) K) 2 ( " P ) (tf-1 • tf-) n
Z-P Q=l
Q "P+«
0 = 1 7=1
x[c1(P,g,l)(e?-1.^)(e^-1.ur^) min{Q,k} UA
= 0 A=l
t=l
/or 1 < P < / - 1 and 1 < a < n'p. To proceed further, we now exploit the integrability condition ddu = 0, which implies for any C2 map u that m—1
ej • u7 =
ei
• u] - E
m' — 1 e
ep(t i> e,-])uj[ -
p=0
E
< ([e'a> e fl]) u °"f •
a,0=O
From this identity we obtain for instance Pa
P-l nQ n'p- „ . . „,P« _ V ^ V " Y ^ ^ ?; <5/5 ? ,(^-0)-. Qf>(P-Qhuo Ai Q=l/3=1 7=1
Taking differentiation by eo successively and invoking Corollary 3.2 with r = I — 1, we then deduce Lemma 3.5. Let u : M —> M ' 6e a proper harmonic map from a k-step Carnot space M to an I-step Carnot space M', which maps N x JL|_ to N' x R+. Suppose that u extends to a Cl map up to the boundaries N x {0} and N' x {0}, and satisfies (3.6) on N x {0}. Then, for each 2 < P < I
194
and 1 < A < min{P — l,k}, the following holds on N x {0}. n
p_A
n
Q
P-Q
Z^ Zs Zs "QfsiP-Q)-, Q=l /3=1 7 = 1
x c2(P,Q,A)(e^1 where c2{P,Q,A)
P-A-V - .p _ ^ _
• u^)(%-A-Q
• uATQh
..
Relabel the indices of the identity in Lemma 3.5 and substitute it back into the identity in Corollary 3.3. Then we multiply the resulting equation by ep~x • uPa and sum it up over a, 1 < a < n'P, to yield
A=l l-P
n'p
n'Q
a=l
n'p+Q
Q = l a = l /3=1 7 = 1
l-P
min{Q,k}
P+Q-A
rip
n'Q n'p+Q
n'R
n'p+Q_R
nA
+E E E EEEE E E Q=l
A=l
R=l
a=l/3=l 7=1
/J=1
"=1
i=l
cUP,Q,A)c 2 (P + Q ) i ? ^ ) & ^ & ^ Q _ f i ) ^ K ) 2 ( ( - p - C ? ) x ( e r 1 • tf-) ( e ? " 1 • &)(<#-* = 0.
• uQ/M+Q'A~R
•
«$J+°-JR)-)
Observing that the summands of the second term are anti-symmetric in P and Q, it is not hard to see that the sum of the second term over P, 1 < P < Z — l , i s 0 . Similarly, since d(P, S - P, A)c2(5, R, A) = (P - l)\(S - A)-1
c3(S,P,A)c3(S,R,A),
where c3(S,P,A)
=
{(P-l)\(S-P-A)\}-\
it is also verified that the sum of the third term over P, 1 < P < I - 1,
195
yields I min{S-l,fc} n's UA
C-D'E E S=2
A=l
EE^-^K) 2 "^
7=1 i = l
x{i:\(S>P>A)£Bf,6j:(s_p)fl(^-1.<0(eoS-p-A-tx2-P,/,)}: <*=1 0 = 1
P=l
Hence we obtain on iV x {0}
Ei^-Dir^E^+^K) 2 ^!:^- 1 -^) 2 P=l ;
A=l min{S—1,*} n's n^
a=l
+ E E EE^rW"^ S=2
A=l
7=1 i = l
^bfLls_^(^.^-)(4-p-A-u%-p^)}i
x {^c3(S,P,A)± = 0.
Since all terms in the sum are nonnegative and u° ^ 0 by assumption, it follows that u satisfies o n J V x {0} ep~l
-up«=0
for each 1 < P < I — 1 and 1 < a < n'p, and
Q=l
a=l
0=1
for each 2 < P < Z , l < y l < min{P - 1, A;}, where 1 < 7 < rip, 1 < i < n A Noting that c2(P,Q,A) = (P - A - l)\c3(P,Q,A) and substituting back into Lemma 3.5 then yields ep~A
• up:
= 0
for 2 < P < I and 1 < A < min{P — 1,/c}. Summing up all these results, we obtain the following necessary conditions for the boundary value of a proper harmonic map between Carnot spaces. Theorem 3.1. Let u : M —>• M' be a proper harmonic map from a kstep Carnot space M to an I-step Carnot space M', which maps N x R+ to N' x R+. Suppose that u extends to a Cl map up to the boundaries N x {0}
196
and N' x {0}, and satisfies (3.6) on N x {0}. Then the following hold on N x {0}. (1) For each 1 < P < I, er0 • UQ" = 0,
0 < r < P - 1, 1 < a < n'P.
(2) For each 2
e0•
uPAa.
min{P - 1, k},
= 0,
0
(3) UQ satisfies the polynomial equation k
min{fc,;}
a
np
n'p
( E H W - E ^{£E(^)2}K)2(/-p) = o. A=l
P=\
i=\ 0=1
Now, we make the following definition related to the nondegeneracy of boundary values. Definition 3.1. Let u : M —)• M' be a proper C°° map from a k-step Carnot space to an I-step Carnot space M', where k > I. Suppose that ueC°°(N
xR+,N'
x J R + ) n C 1 ( A f x [0,oo), N'x
[0,oo))
l
and the boundary value f € C (N x {0}, N' x {0}). Then we say that f is nondegenerate if it satisfies dfP((ni)P) <jL Y,(n'j)f(P) at any p € N x {0}, which is equivalent to assuming (3.4). If/ = 1 in particular, then with the convention n'0 = {0}, this condition simply means that df ^ 0 everywhere on TV x {0}. Note that when M and M' are real hyperbolic spaces, this is equivalent to the condition e(/) > 0 assumed in Theorem 1.1. As a notable special case of Theorem 3.1, we finally deduce Corollary 3.4. Let k > I. Letu : M —> M' be a proper harmonic map from a k-step Carnot space M to an I-step Carnot space M', which maps N x R+ to N' x R+. Suppose that u extends to a Cl map up to the boundaries N x {0} and N' x {0}, and the boundary value f of u is nondegenerate. Then the following hold on N x {0}. (1) For each 1 < P < I, erQ • u^ (2) For each er0 • UA) = 0,
= 0,
0 < r < P - 1, 1 < a < n'P.
2
197
(3) UQ satisfies the polynomial equation I
nP
n'p
P=l
i=l
0=1
k
A=l
Prom Corollary 3.4 (2), considering the case with r = 0, we see that the harmonic map u must satisfy for each 1 < A < P — 1 and A + 1 < P < I u^a. = 0 o n i V x {0}, which implies that / = u\N x {0} satisfies for each 1 < i < I
*fe)p)cEK)/(p) j=i
(3-7)
j=i
at anyp £ Nx {0}. This means that in the case where k > I and X — n = 1, the nondegenerate boundary value of a proper harmonic map from a A;-step Carnot space to an /-step Carnot space, having Cl regularity up to the boundary, must preserve the filtrations on the boundaries / ; ni C nx + n 2 C • • • C ] P n j ; ni C ni + n^ C • • • C ^ n ^ - , where rij and nj are the distributions defined by the graded Lie algebra decompositions n — 5Z i = 1 nj and n' = ^ . = 1 n!,, respectively. Since at the boundary the quantities Up. for 1 < i < np and 1 < /? < n'P are determined by the boundary value / of u, denoting Up. by fp13, we obtain from Corollary 3.4 (3) the following equation k
I —I
np
A=l
P=l
i = l /3=1
n'p
ri;
n\
i=l 0=1
as a necessary condition satisfied by / on Z . V x {0}. Since n,
n\
EE(^f)2>° 8 = 1 (3 = 1
by the nondegeneracy condition of / , this equation has a unique positive solution UQ > 0. Namely, in the case where k > I and A — fi = 1, for a given proper harmonic map u with nondegenerate boundary value / , the ej
198 v be two proper harmonic maps satisfying the conditions in Corollary 3.4. Assume that u and v have the same nondegenerate boundary value / on TV x {0} and d,M'{u(p),v(p)) -> 0 as p ->• oo, where djvf(u(p),u(p)) is the distance between u{p) and v(p) in M', and oo denotes the point at infinity defined by R+ directions in TV x R+ (cf. Proposition 2.2). Then u and v are identical. As in the case of real hyperbolic spaces remarked in Subsection 1.3, for proper harmonic maps between Carnot spaces which do not have sufficient regularity up to the boundary, the uniqueness does not hold in general. In fact, for arbitrary fc-step Carnot space M, Watabe [8] constructed a one-parameter family of harmonic diffeomorphisms of M which induce the identity map on the ideal boundary. Finally, we remark that, with appropriate modifications, Corollary 3.4 and the filtration preserving property (3.7) also hold in the general case where k > I and A = \i £ N are arbitrary integers > 1. For the details we refer the reader to [6]. R e m a r k 3 . 1 . In the case where k < I, Corollary 3.4 (3) claims only that k
O\2(i-*0
K)
k
nP
rip
(E^)K)"-E^{EE(/?)a}K)5
X1(k-P)
J Pi ) / l " 0 ,
. A=l
P=l
i=l
0.
0=1
Consequently, when UQ = 0 at the boundary, we do not in general have good control of UQ in terms of the boundary value f, while at the points where UQ ^ 0 on the boundary, we can reconstruct UQ from f provided YM~\ Z)/3=i(Af) > 0; since the equation above reduces to k
2t
k—\
f
np
rip
2
(EH«) -E {EE(/?) }K) A=l
P=l
i = l 0=1
2M,
nk
n'k
i=l
0=1
-^E(/?)2 = o.
3.3. The case of A =£ fj, The asymptotic behavior of a proper harmonic map between Carnot spaces becomes more subtle when A / p., that is, in the case where the pullback metrics $*# and $'*#' in (3.2) on the half space models N x R+ and TV' x R+ blow up with different growth orders at the boundaries A^ x {0} and N' x {0}, respectively. Consider again a proper C°° map u : M -> M' from a /c-step Carnot space M to an /-step Carnot space M', which maps N x R+ to N' x R+. We suppose that u extends to the boundaries N x {0} and N' x {0} with
199
sufficient regularity, and plan to investigate necessary conditions satisfied by the boundary value of u, when u is harmonic in the interior M. First, by applying the same argument that derives Lemma 3.1, we obtain the following result concerning the asymptotic behavior of the tension field T(U) of u. Lemma 3.6. Let u : M —> M' be a proper C°° map from a k-step Carnot space M to an l-step Carnot space M', which maps N x R+ to N' x R+. Suppose that u extends to a C1 map up to the boundaries N x {0} and N1 x {0}. Then, for each 1 < B < fil, with respect to adapted frame fields, the local expression T(U)° in (3.3) of the tension field T(U) of U satisfies the following. (1) As y —> 0, the first three terms ofr(u)0 x (y' ou)2,il~1y~2B converges to 0 if B < fil, and to
-(A£W)K)2M'
ifB = id.
(2) The fourth term of T(U)° X (y' o u ) 2 M ' _ 1 y _ 2 B is given by
0=1
p
nA
n'p
p
+^EE EE(^^^ ou ) M( ' _p) «2) 2 +°( 1 )' P
A
i = l 0=1
where integers P range over I + /x _ 1 (l — B) < P < I in the first sum and I + /x_1(A — B) < P < I in the second sum, while integers A range over 1 < A^mir^A-^B + M C P - O ^ From this lemma we can deduce, as in the case of A = /x, necessary conditions to be satisfied by the boundary value of a proper harmonic map. Since the situation is rather complicated in full generality, let us now confine our investigation to the most typical cases. Namely, in order to bring out key features, we first examine the cases where k and / are 1 or 2, and kX = In. 3.3.A
The case where k = 1, I = 2 and A = 2/x, n £ N
Suppose in the following that A = 2fi, n £ N, and u is a proper harmonic map which extends to a C 2 M map up to the boundaries. Then from Lemma 3.6 we can deduce the following
200
Proposition 3.4. Let u : M —» M' be a proper harmonic map from a 1-step Carnot space M to a 2-step Carnot space M', which maps N x R+ to N' x R+. Suppose that u extends to a C 2/J map up to the boundaries N x {0} and N' x {0}, and satisfies
EE(%?)2^0
onNx{0}.
(3.8)
i = l 0=1
Then the following hold on N x {0}.
ero-u1oa=0,
0
er0-u^
0
=0,
1 < a
A new feature here is that from the assumption (3.8) we now obtain u° J^ 0 on the boundary, although, in the case where A = fi = 1 and k < I, as observed in Subsection 3.2, we can not derive it from the assumption (3.4), that is, from a condition that depends merely on the differential of the boundary value. In order to derive further necessary conditions, we now consider other components r(u)Pa of the tension field T(U). Then, corresponding to Lemma 3.4, we can easily deduce Lemma 3.7. Let u : M —> M' be a proper C°° map from a 1-step Carnot space M to a 2-step Carnot space M', which maps N x R+ to N1 x R+. Suppose that u extends to a C 2 M map up to the boundaries N x {0} and N' x {0}, and satisfies UQ ^ 0 on N x {0}. Then, as y -* 0, the local expression T(U)P" in (3.3) of the tension field T(U) of u satisfies the following. (1) For 1 < P < 2, the first three terms of T(u)Pa x (y' o u)4/t-/iP-ly-4^+l
converges
t0
-{( M P - 1)!}-I(2n1 + P ^ K ) (2) For P = 1, the forth term of verges to
{(/i - i)i(2M - I)!}-1 K r
1
4
^-
T{U)P<* X
1
(ef-1
• <-).
(y1 oU)4'i-'iP-1y-^+1
con-
£ Y, £ i , K-1 • «S') {4"-1 • <")• /3=17=1
It is immediate from this Lemma that the C 2 M boundary value of a proper harmonic map satisfies the following conditions, since r(u)Pa = 0.
201
Corollary 3.5. Under the assumption of Lemma 3.7, if u is a harmonic map, then it satisfies on the boundary N x {0}, e
o _ 1 • "o" = °»
e
oM_1 • $
= 0,
l
Summing up all these results, we then obtain the following Theorem 3.2. Let u : M —> M' be a proper harmonic map from a I-step Carnot space M to a 2-step Carnot space M', which maps N x R+ to N' x jR + , and A — 2\i, \x G N in (3.2). Suppose that u extends to a C 2 M map up to the boundaries N x {0} and N' x {0}, and satisfies ni
«2
i = i 0=1
Then the following hold on N x {0}. (1)
(2)
u°0 ± 0.
er0-u10a=0, eg-Uo' 3 =0,
(3) WQ satisfies the
0 < r < / i - 1, 1 < a 0 < r < 2 / / - 1, 1
equation m
n2
4
An1(n8) " = 2 / xX;EWf)i = l (8=1
5.5.5
77ie case where fc = 2, Z = 1 and \i = 2A, A € AT
Suppose now that /z = 2A, A G iV, and u is a proper harmonic map which extends to a C 2A map up to the boundaries. Then in this case we deduce from Lemma 3.6 the following Proposition 3.5. Let u : M —>• M' be a proper harmonic map from a 2-step Carnot space M to a l-step Carnot space M', which maps N x R+ to N' x R+. Suppose that u extends to a C 2A map up to the boundaries N x {0} and N' x {0}, and satisfies
EE(U2?)2^0
onNx{0}.
j=i 0=1
Then the following hold on N x {0}.
er0 • u£" = 0, r 60 •
1/3 U
U
0 < r < 2A - 2,
202
Moreover, by investigating the limit of T(U) 1 Q X (y' o u) 2 A _ 1 y~ 4 A + 1 as y —> 0 and utilizing the integrability condition ddu — 0 as in Subsection 3.2, we can also deduce that under the assumption of Proposition 3.5, u satisfies on the boundary N x {0}, e ^ " 1 . u\p = 0,
e£-11^=0,
l<*
Combining these results, we finally see that the following theorem hold in this case. Theorem 3.3. Let u : M —> M' be a proper harmonic map from a 2-step Carnot space M to a l-step Carnot space M', which maps N x R+ to N' x R+, and y, = 2A, A G N in (3.2). Suppose that u extends to a C 2A map up to the boundaries N x {0} and N1 x {0} ; and satisfies n2
n
'i
EX>2?)V0
onNx{0}.
»=1 /3=1
Then the following hold on N x {0}. (1) u°0 ? 0. (2) e 5 - u j " = 0 , 0 < r < 2A - 1, 1
(E^)(«g) A=l
4A
n2 "i
= 25:E(*)ai-\
/3=1
The details of the proof and results in more general setting will appear elsewhere. Acknowledgments The author would like to thank Keisuke Ueno for his collaboration throughout this work. This work was partly supported by the Grant-in-Aid for Scientific Research (A), No. 10304004, and the Grant-in-Aid for Exploratory Research, No. 12874008, of the Japan Society for the Promotion of Science. References 1. P. Eberlein, Geometry of Nonpositively Curved Manifolds, The University of Chicago Press, Chicago, 1996. 2. H. Donnelly, Dirichlet problem at infinity for harmonic maps: Rank one symmetric spaces, Trans. Amer. Math. Soc. 344, 713-735 (1994).
203 3. E. Heintze, On homogeneous manifolds of negative curvature, Math. Ann. 211, 23-34 (1974). 4. P. Li and L.-F. Tarn, Uniqueness and regularity of proper harmonic maps, Ann. of Math. 137, 167-201 (1993). 5. S. Nishikawa, Harmonic maps and negatively curved homogeneous spaces, Geometry and Topology of Submanifolds X, World Scientific, Singapore, 200215, 2000. 6. S. Nishikawa and K. Ueno, Asymptotic behavior of proper harmonic maps between Carnot spaces, in preparation. 7. K. Ueno, Non-existence of proper harmonic maps from complex hyperbolic spaces into real hyperbolic spaces, Tohoku Math. Publ. 20, 189-195 (2001). 8. D. Watabe, Dirichlet Problem at Infinity for Harmonic Maps, Tohoku Math. Publ. 18, 71pp. (2000).
H A R M O N I C COHOMOLOGY GROUPS ON COMPACT SYMPLECTIC NILMANIFOLDS
YUSUKE SAKANE, TAKUMI YAMADA Department of Mathematics, Osaka University, Toyonaka, Osaka 560-0043, JAPAN In this paper we summarize recent results on harmonic cohomology groups of compact symplectic nilmanifolds. We also study harmonic cohomology groups of dimension 3 and of codimension 2 for 2-step nilmanifolds and give examples of compact 2-step symplectic nilmanifolds G/T such that dimension of harmonic cohomology groups varies.
1. Introduction Let (M, ui) be a 2m-dimensional symplectic manifold and let flk(M) denote the space of all fc-forms on M. As an analogy for an oriented Riemannian manifold, Brylinski [B] denned the star operator * : flk(M) ->• Q,2m~k(M) for a symplectic manifold and the operator d* = (—l)k * d* on Q,k{M). He also introduced the concept of symplectic harmonic forms. A form a is called harmonic form if it satisfies da = d*a = 0. Let Ti^M) denotes the space of all harmonic /c-form and we define symplectic harmonic fc-cohomology group Hk_hr (M) of the symplectic manifold (M, w) by nk(M)/(Bk(M) n nk{M)), which is a subspace of de Rham cohomology group HpR(M). Brylinski conjectured that, for a compact symplectic manifold, every de Rham cohomology class contains a harmonic representative. He proved that this conjecture is true for compact Kahler manifolds. We denote by La : Qk(M) ->• Qk+2(M) the multiplication by u and by L[w] : HpR(M) -> H^R2(M) the induced homomorphism in de Rham cohomology groups. However, Mathieu [Ma] proved the following theorem. Mathieu's Theorem. Let (M 2 m ,w) be a symplectic manifold of dimension 2m. Then following two assertions are equivalent: (a) For any k < m, the homomorphism L^k : H^Rk(M) ->• H^k(M) is surjective. (b) Any de Rham cohomology class contains a harmonic cocycle, that is, for any k, HkDR(M) = Hk_hr(M) Mathieu's theorem is a generalization of Hard Lefschetz Theorem for compact Kahler manifolds. Yan ([Yn]) gave a simpler, nice proof of Mathieu's Theorem by using a special type of infinite dimensional s((2)-representation theory. Related with the study of harmonic forms, we are interested in the following question raised by B. Khesin and D. McDuff (see Yan [Yn] ).
204
205
Question : On which compact manifold M, does there exists a family u)t of symplectic form such that the dimension of Hk_hr(M) varies for some Jfc? This question was considered by Yan [Yn] for 4-manifolds and he constructed compact 4-manifolds M which have a family wt of symplectic forms such that the dimension of Br\ _hr{M) varies. Yan also observed that for compact 4-dimensional nilmanifolds the dimension of H^ _hr{M) is independent of symplectic forms. For higher dimensional compact nilmanifolds, the question above is considered by Yamada [Ym] and Ibanez, Rudyak, Tralle and Ugarte [IRTU] independently. Ibanez, Rudyak, Tralle and Ugarte [IRTU] proved that there exit at least five 6-dimensional nilmanifolds M with a family ut of symplectic form such that the dimension of Hk _hr(M) varies, by computing harmonic cohomology groups H* _hr(M) and Hi _hr{M). such that Note that if there exist a diffeomorphism / : Mi —> M^ between symplectic manifolds (M\,UJ\) and (M.2^2) such that f*(u>2) = wi, then f"(H^.hr(M)) = H^.hr(M). In Yamada [Ym] it is proved that if M is a compact manifold and u>, ui' are symplectic forms on M such that ui — UJ1 = dry, then Hk_hr(M) = H*,_hr(M), that is, the subspace Hk_hr(M) oiH%R(M) does not depend on the choice of symplectic form u> of the de Rham class [UJ]. From this result, together with a theorem of Nomizu on compact nilmanifolds, we may assume that symplectic forms on compact nilmaifolds are left invariant and it is enough to consider subspaces of Lie algebra cohomology groups to study harmonic cohomology groups of compact nilmaifolds. In section 3, we study harmonic cohomology groups of compact nilmaifolds of 3-dimension and codimension 2. In section 4, we give examples of compact 2-step symplectic nilmanifolds G/T such that dimension of harmonic cohomology groups varies. 2. Operators on fi*(M) and harmonic cohomology group A Poisson manifold is a smooth manifold M endowed with a Poisson structure G, that is, a skew-symmetric contravariant 2-tensor G on M satisfying [G, G] = 0, where [ , ] denotes the Schouten-Nijenhuis bracket. Let flk(M) denote the space of all fc-forms on M. For a Poisson manifold M, Koszul [K] introduced a differential operator d* : Ofc(M) ->• flk~1(M) defined by d* = [d,i(G)},
(1)
where i(G) denotes the contraction by G. The operator d* is called the Koszul differential. Let (M, UJ) be a 2m-dimensional symplectic manifold and let G be a skew-symmetric bivector field dual to UJ. Then G is a Poisson structure on M. By the Darboux's theorem, we can write in canonical
206
coordinates w = dpi A dqi + ... + dpm A dqm and d 8 d d dqi dpi dqm dpm Brylinski [B] defined the star operator * on a symplectic manifold (M,u) so that the Koszul differential is a symplectic codifferential of the exterior differential d with respect to the star operator. We choose the 2m-form vM = w m / m ! as a volume form on M. Then we define the star operator * : fi*(M) -»• n 2m - fc (Af) by the condition a A * /? = (AkG)(a,/3)vM for all a,P € flk(M). operator satisfies the identities *2 = id,
d* =
The star
{-l)k*d*.
Yan [Yn] introduced the operator L* : fi*(M) -> ftk~2(M) as the symplectic adjoint operator of L — Lu, that is, L* — *L*. Then we see that i(G) = — L*. Furthermore, the operators L, d, d* and L* acting on the algebra fi*(M) satisfy the following commutator relations: [L, d) = 0, [L, d*] = d, [L*, d*) = 0 and [£,*, d]=d*.
(2)
Note that for a symplectic manifold, we have dd* + d*d = 0 on fl*(M). Definition 2 . 1 . For a symplectic manifold (M,ui), a fc-form a on M is called w-harmonic, or simply, harmonic if it satisfies da = d*a = 0. We denote the space of all w-harmonic fc-forms by %* (M), or simply, Hk(M). Definition 2.2. We define harmonic fc-cohomology group Hk_hr(M) of the symplectic manifold (M,w) by Hk(M)/(Bk(M) flKj(M)), as a subspace of de Rham cohomology group HpR(M). For simplicity, we also denote harmonic fc-cohomology group by Hfcr(M). Now the operators L and L* acting on the algebra fi*(M) also satisfy the relation [£,*, L]a = (m — k)a for all fc-form a (see Yan [Yn]). We define an operator A on tt*(M) by 2m
A = ] P ( m - k)nk where 7Tfc : fi*(M) ->• f2A(M) is the projection. These operators satisfy the following relations: [L*,L] = A,
[A,L] = -2L,
[A, L*} = 2L*.
(3)
We now consider the Lie algebra sl(2) with the standard basis H, X, Y: [X,Y] = H,
[H,Y] = -2Y,
[H,X] = 2X.
(4)
207
Definition 2.3. Let V be a vector space (possibly infinite dimensional) and assume that the Lie algebra sl(2) acts on V. We say that V is an sl(2)module of finite if-spectrum if the following two conditions are satisfied: (1) V can be decomposed as the direct sum of eigenspaces of H. (2) H has only finitely many distinct eigenvalues. Let Vk be the eigenspace of H with respect to eigenvalue k. Then we have the following Lemma. Lemma 2.1 ([Yn]). Let V be an sl(2) -module of finite H-spectrum. the linear mapping Yk:Vk^V-k
and
Then
Xk:V-k->Vk
are isomorphisms. Furthermore, let Pk = {v€Vk\Xv
=ti}= {v£Vk\
Yk+lv
= 0}
(5)
be the set of primitive elements in Vk. Then we have the decompositions: V_k=YkPk®Yk+1Pk+2®---.
Vk=Pk®lm(Y),
(6)
Now we have a representation of s[(2) on fl*(M) by sending X —> *L*,
Y —> L,
H —> A.
It is easily see that fi*(M) is an s[(2)-module of finite iJ-spectrum. Thus we have the following. Proposition 2.2 ([Yn]). (Duality on forms) The linear mapping Lk : Q,m~k(M) —>• Qm+k(M)
is an isomorphism.
Proposition 2.3 ([Yn]). (Duality on harmonic forms ) The linear mapping Lk
. ^ m - f c ( M ) _>. ^ r n + k ^
{s
Qn
isomorphism.
In particular, we have H™r+k{M) = Im{Lk : H%Tk(M) -> H%+k(M)}. We remark that Yan [Yn] gave an alternative proof of Mathieu's Theorem in Introduction by using Lemma 2.1 and Proposition 2.3. Note also that we have Hlhr(M) = HlDR{M) for i = 0,1,2. Thus we have the following from Proposition 2.3. Corollary 2.4. We have H2™-\M)
= ImfL™- 1 : HlDR{M) - • H2DmR-\M)}
(7)
H2hr\M)
= Im{L™-^: H2DR(M) -+ H2DmR~2(M)}.
(8)
and
Using Corollary 2.4, Ibaiiez, Rudyak, Tralle and Ugarte [IRTU] computed harmonic cohomology groups H*_hr (M) and H^_hr (M) for 6-dimensional compact symplectic nilmanifolds M.
208
Now the following observation due to Yan is a key lemma to prove that the subspace H^_hr(M) of H^R(M) does not depend on the choice of symplectic form UJ of the de Rham class [w] Lemma 2.5 ([Yn]). Let (M,w) be a symplectic manifold. PutP™~k(M) {veH%Rk(M)\L^v = 0}. Then
=
P™-k(M)cH™h*(M). Now we get following from Lemma 2.5, Lemma 2.1 and Proposition 2.3. Proposition 2.6 ([Ym ]). Let M be a compact manifold and plectic forms on M such that UJ — ui' = dry. Then we have HlhT{M)=Hl,_hr{M)
forq =
OJ,UJ'
sym-
0,...,2m.
3. Harmonic cohomology group of nilmanifolds Now we consider the case of compact nilmanifolds. Let g be a Lie algebra and put g(°) = g and let g( l+1 ) = [g,gM] for i > 0. We say that a Lie algebra g is (r + l)-step nilpotent if g(r) ^ (0) and g^r+1^ = (0). We say a Lie group G is (r + l)-step nilpotent if the Lie algebra g is (r + l)-step nilpotent. If G is a simply connected (r + l)-step nilpotent Lie group and r is a lattice of G, that is, a discrete subgroup of G such that G/T is compact, then we say G/T is (r + l)-step compact nilmanifold. Note that, by Malcev's theorem, G admits a lattice if and only if g admits a basis such that all the structural constants are rational. We denote by /\ g* the space of all left G-invariant fc-forms on G and regard it as a subspace of Clk(G/T). Then we have a subcomplex (f\*Q*,d) of the de Rham complex (fT(G/r),d) of compact nilmanifold G/T and denote by Hh(g) the fc-th cohomology groups of the complex (/\*g*,d). Theorem 3.1 (Nomizu). For a compact nilmanifold G/T, the natural inclusion L : (/\*g*,d) -> (f2*(G/r),d) induces an isomorphism on cohomology groups i*:Hk(g)=HkDR(G/T). For a symplectic form CJ on a compact nilmanifold G/T, there exists a G-invariant closed 2-form w0 on G/T such that u> — u>o = d-y. Note that UJQ is also a symplectic form on G/T. For a G-invariant symplectic form LOQ, we denote by %k(o) the space of all G-invariant harmonic fc-forms on G/T. Lemma 3.2 ([Ym]). Letuj0 be a G-invariant symplectic form on a compact nilmanifold G/T. Then we have Lk:Hm-k{9)^Hm+k(g) is an isomorphism for any k.
209
For a G-invariant symplectic form wo, let
Ht0.hr(a) = nk(B)/(Bk{B)nnk(a)) be a subspace of the cohomology group Hk (g). Proposition 3.3 ([Ym]). Let (G/T,ui) be a compact symplectic nilmanifold and let wo be a G-invariant symplectic form such that LO — UIQ = dj as above. Then we have HlhT{GlY)
= Ht0.hr(G/T)
= HkUo_hr(Q)
for any k. From now on we assume that symplectic structures on a compact nilmanifold G/T are G-invariant. At first we consider a relation of symplectic harmonic cohomology groups Hlhr(M) = H}jR(M) and H2™-l{M) for a compact nilmanifold M2m = g y p Theorem 3.4 ([Ym]). Let ( M 2 m = G/T,w) symplectic nilmanifold. Then we have
be a (r + l)-step compact
dim HJoniM) - dim H2h™-l{M) > dimgW. In particular, if M is a 2-step nilmanifold, dim HlDR(M)
- dim H2^1
(M) = dim[g, g].
(9)
Let g be an (r + l)-step nilpotent Lie algebra. Consider the descending central series { g ^ } of g, where g^ +1 ) = [g,g^] and g(°' = g. Let a ^ denote the complement vector subspace of g( l+1 ) in g ^ : fl(0 = fl(*+D + 0 (0 for i = 0 , 1 , . . . , r — 1 and let ni = dim a ^ . For simplicity, put /\10'" 'lr = A io o(°)* A . . . A A i r a W *- Then we have
A'»- = E„ + ... + _A" -• Now we have the following Lemma due to Benson and Gordon [BG]. Lemma 3.5 ([BG]). Each closed 2-form 0 £ A20* belongs to f\ ' ' "' ' +
EA io,.-. ,i,— l ,0 Let {Ai,... , Anr,} be a basis of /\ ' "' ' . By Lemma 3.5, a G-invariant symplectic form w can be written in the form u) — Pi A Ai + . . . + pnr A A„r
modulo
/.IX
> (10)
where /?i,... ,/?„r are elements of [\ ' ' "' . Since ui is non-degenerate, we see that j3i,... ,/3nr are linearly independent. We extend these elements to
210
a basis {Pi,... ,/?„„,... ,/3no} for / y 1 ' 0 - - 0 . p u t 0 J 0) = span{/?i,... ,/?„„} and 4 0 ) * = s p a n ^ + i , . . . , / 3 n J . Then 0(°>* = oj 0) * +<4 0) *. Put nj = dim a[' = nr and TIQ = dim c4 = n0 - nr. For simplicity, we also put A (iJ,i§),M,...^ = ^ ( o ) * A A i? a (o)* A A i l a ( 1 ) * A _ A A i . a W * _ Moreover, let {/?{ ,... ,Pn,k} be a basis of a^'* and put {Wi, . . . ,U>2m} = {Pi, • • • , Ai0> •• • >Pl i • • • i^nfc > • • • i ^1 > • • • > V } Let { X i , . . . , X2m} be a basis of g which is dual to the basis {cui,... , W2m}If we write down a symplectic form w as w = ^Ja^-Wi A Wj
ay = —a^ G ffi,
then it is easy to see that the Poisson structure G dual to u> is given by G = -^cijXiAXjJ
(11)
where (c^) is inverse of transpose matrix of (a^)- Thus we have the following. L e m m a 3.6. With respect to the basis {X\,... form /
^nr,na—nr
^nr}nr
(cij) =
vnr,2m—no—nr
* * *
^no—nr,nr
^2m-no-nr,nr
V -Enr
,X2m},
* * *
G is given of the
EnA * *
(12)
* /
L e m m a 3.7. Let u = /?iAAi + . • -+/3 n i AAni modulo /\ ' be a G-invariant symplectic form on 2-step nilmanifold G/T, then we have x-^
u
x-^
u
A (2,0),0
A
(1,1),0
Corollary 3.8 ([SY]). If (G/T,co) is a 2-step symplectic nilmanifolds, then we have B3(g) CH3(g). ,3,0
2,0
Proof. Since d/\ 0 and df\ 1,1 ' C A, ' , we only consider the case of , 0,2 /\ ' . From Lemma 3.7, we see that . 0,2
J
. (2,0),1
A ^A
A
(1,1),1
+A •
Thus, from (12) in Lemma 3.6, we get that
i(G)(d/\°'2) c f\ which implies that d/\ 0 ' 2 C W3(fl)
•
211
Proposition 3.9 ([SY]). If M2m = G/T is a 2-step nilmanifols, then we have C U2m-q{Q).
(13)
In particular, we have dim H2DR{M)A
dim H2h™~2(M) =
no—3,«i
„
o
A
no—1,n\—l
nn2m-2(&))+dimdf\
dim(d/\
-m.
(14)
Proof. Prom (12) in Lemma 3.6, we have A
no,"i —a
A
. (nn,"n),ni — a
') =i(G)(/\(
i(G)(/\ n
( o>"o
_2
n
)> i~9
')
A "o-l,"i-9-l
+A
A
no,"1-9-2
+A
Thus we have di{G){/\n°'ni~q) = 0 from Lemma 3.7. Note that d/\n°'ni~9 = 0. Since d* = [d,i{G)}, we see that /\ n °> ni -« c H 2m -«(fl). To prove the second assertion, note that d/\n°~ 'ni~ is a subspace of 2m 2 ^ - ( g ) , since df\n"~2^-1 C / p ' " 1 " 2 . Thus we have dim^r(fl)-dimH^-2(fl) = dim ft2 ( 0 ) - dim(B 2 ( fl ) n H2(g)) - d i m f t 2 m - 2 ( 0 ) + dim(B 2 r o - 2 ( 0 ) n
n2m~2{g))
dim(B2m-2(S)nn2m-2(e))-dimB2{Q)
=
A A
no —3,ni
o n
A no—2-m—1
no — 3,ni
nH 2 m„ - 2 (fl)) „
+ d i m d»/n\o — 2 , n i — 1
nH 2 m - 2 ( 0 )) + dimd/\
„
-dimS2(g)
-m.
D
4. Examples Now we give examples of compact 2-step symplectic nilmanifolds G/T such that dimension of harmonic cohomology groups varies. Example 4.1 ([Ym], [IRTU]). Consider a 2-step nilpotent Lie algebra g of dimension 6 given by 0=
span{X1,X2,X3,Xi,X5,X6}
with [Xi,X2] = XQ, [X2,XS]
= Xi, [X$,X\] = X5.
212
Let {co>i,W2,W3,W4,w5,u;6} be the dual basis of {XI,X2,X3,X4,X5,XQ}. Then the space of invariant closed 2-forms Z2(g) is given by (
Z2 (fl) = < 5Z aii UJi A w-?
0 -a12 — 013
a12 ai3 0 a23 —023 0
«25 + 036 -024 - ^ 3 4 -015 - a 2 5 -035 \ -Oi6 - a 2 6 -036
i
- a 2 5 - o36 a15 a16\ a24 025 026 034 035 036
0 0 0
0 0 0
0 0 0/_
and the space of all G-invariant symplectic forms iS(jj) are given by 016034025 — 034026015 + 02403601s'
5(fl) = {
+ 0 3 6 0 2 5 - 0i603 5 024 - 025035026 2 + a 2 5 0 3 6 - 035026036 ^ 0
Let -4(g) be a subset of 5(g) defined by
= {^2 aii Wi A ^i e
A{Q)
O15O24+O25 + Oi 6 034
S
^
+035026+025036 + 0 3 6 = 0
and let B(g) be a subset of <S(g) defined by
Bid) - I ^2 aiiWi
AW
J
025 = O36 = — ^ O i 4 ^ 0 , O i 5 , O i 6 £
e S
(& 024 = 026 = O34 = 035 = 0
i<j
U {^2 aijuji
AW G5 0
J
()
j
014 = O36 = ~ « 0 2 5 7^ 0 , O 2 4 , O 2 6 £ 015 = Oi6 = O34 = 035 — 0
1
U 1 53 aii Wi A ui
Ol4 — O25
GS
^> 015 = «16 =
a
036 7^ 0 , 0 3 4 , 0 3 5 €
2 4 — O26 — 0
Proposition 4.1. For a compact symplectic nilmanifold G/T with Lie algebra g, we have dimHl(G/T)=0, dimH2DR(G/T) 2
dimH
DR(G/T)
- dimH4hr(G/T) = 1 4
- dim H hr{G/T) = 0
for to £ .4(g), for w £ 5(fl) - ^(g),
and d i m i ^ r ( G / T ) = 10 dimi^r(G/r) = 9
for u £ 23(g), /or a; £ S(g) - B(Q).
213
Note that dim HhR(G/T)
= dim H5DR(G/T)
= 3,
dimHlR(G/T)
= dim H^G/T)
= 8,
dimHf}R(G/T)
= 12.
Example 4.2. For p > 2 let f)(l,p) be a 2-step nilpotent Lie algebra of dimension 2p + 1 spanned by {Xi,... ,X 2 p + i} with [Xi,Xi]=
Xp+i
i=
2,...,p,
We consider the Lie algebra cj = fj(l,p)©K of dimension 2p + 2. Let X 2 p + 2 denote a generator of the Lie algebra M. and let {wi,... , <x>2p+i, w2p+2} be the dual basis of {Xi,... , X 2 p + 1 , X 2 p + 2 } . Then we have * (o,i),o A (o,o),l l\ D span{w 2 p + 2 }, f\ = s p a n { w p + 2 , . . . ,w 2p +i}Consider a (7-invariant symplectic structure u. We write the Poisson structure G as G = —'Y^cijXiAX; with respect to the basis {Xi,... , X2p+i, X2p+2 above. Then we have the following. Proposition 4.2 ([SY]). For a compact symplectic nilmanifold G/T with Lie algebra g, we have dimH2DR(g)-dimH2hl(g)
PC2,
if Ci2p+2 ^ 0
= \P I P C2 + (p- 1) ifc12p+2 = 0 . To prove Proposition 4.2, we use the relation (14) of Proposition 3.9, the expression of the Poisson structure G and the operator d* — [d,i(G)]. Example 4.3 ([SY]). We consider a 2-step nilpotent Lie algebra of dimension 2m given by g = span{Xi,... ,Xm,Yi,...
,Ym}
with [Xi,Xi+i] = Yi i = 1 , . . . ,m, where we set Xm+i = X\. Then a simply connected nilpotent Lie group G with the Lie algebra g has a lattice Y. Let {/?i,... , /3 m , A i , . . . , Am} be the dual basis of {Xi,... ,Xm,Yi,... ,Ym). Note that df\ ' = (0) and d/\kj C [\k+2'i-1. it i s easy to see that d : / \ 0 j -» f\2'3"l is injective. The space of invariant closed 2-forms Z2 (g) is given by 9
(\—>m
Z2{g) = \ )
i^—^i=i
v—^ m
Oii+i(ftAAi+i-/5i+2AAj) + >
where we put / ? m + i = /?i,/? m + 2 = ,fl 2 ,A m+1 = Ai.
*-—Ji~\
aiipiA\i
+
214
Now we consider a symplectic form u given by an element of /\ ' . Then the Poisson structure G is of the form
E with respect to the basis 3,0
m . . , Cij x i A Yj «,i=i
, F m } . We see that
{Xi,.
-»(0),
i(G):/\
2,1
0,1
1,0
->/\
,
»(G):/\
-> / \
and the space of harmonic 3-forms %3(fl) is given by
«3(e) = A 3,0 + z 3 (s) n A 2 ' 1 + n *w n A1,2For m > 6, we see that
^ 3 (fl) n A
= { Y,j=1 b> &+1 A A* A A i + ! I &j e R, j = l, • ,mj
where we we put fim+i = /3i, A m + i = Ax. Case 1. The case when w = J2T=iajj Pj A -\? where a,jj ^ 0. V—^m
1
Note that G = - ^
Q
A 1'^
— X,- A Y, and hence we have W (fl) n j \ •?=1
= (0)
a
jj
for m > 6. Case 2. The case when u = Y^T=i ajj+i(Pj A ^j+i ~ Pj+2 A Aj). Note that w is given by a form I (
°
0 —ai2
A=
112
0
0 0
123
0 0 0
— 0.23
0 0
0 0 0
0 0 0 0
Vo
0 0
0 0
t
0
••
0
••
«34
0 •• Q45 • •
-«34
0 0
0
—<J45
0 0 0 0
0 0 0 0
.
0 •• 0
••
0 •• 0 •• 0 •• • — 0
••
1 for the basis above, where 0 0 0 0 0 0
0 0 0.m-3m-2
0
—a m— 0 0 0 0 0
0 0 0 0 0 0
Q-m
1 m
0
-3m-2
0 0
a
m
-am - 2 m - l
-2 m - 1
0 0
0 0 0 0 0 0
\
0 0 G-m — 1 m
0
We can see that, for m > 3, the matrix A is non-degenerate if and only if m = 3£ for I G N. Put A'1 = (cfj-) for m - U. Then we see that the components of the matrix A - 1 = (c^) satisfy the conditions Cjj = 0 for j = 1, • • • , m,
Cj+ij = 0 for j = 1, • • • , m — 1 and Cim = 0.
( 0 —A" Since the poisson structure G is given by I t ._1
, we see that, for
a — J^jLi^j Pj+i A Xj A Aj+i, i(G)a = 0 and hence, a is harmonic. This implies that H3(Q) n A*' = ^ 3 (fl) n A*' a n d d i m ( ^ 3 ( B ) n A ) = 3 £ f o r £ > 2. Thus, from Corollary 3.8, we see that compact 2-step nilmanifolds
215
G/T admit symplectic structures such that dimension of harmonic cohomology group H^G/T) varies. References [B] J.L. Brylinski : A differential complex for Poisson Manifold, J. Differential Geometry 28 (1988), 93 - 114. [BG] C. Benson and C. Gorgon : Kahler and Symplectic structures on nilmanifold, Topology 27 (1988), 513 - 518. [IRTU] R. Ibahez, YU. Rudak, A. Tralle, L. Ugarte : On symplectically harmonic forms on six-dimensional nilmanifolds, Comment. Math. Helv. 76 (2001), 8 9 - 109. [K] J.-L. Koszul : Crochet de Schouten-Nijenhuis et cohomologie, in Elie Cartan et les mathematiques d'aujourd'hui, Soc. Math, de Prance, Asterisque, hors serie (1985), 257 - 271. [M] O. Mathieu : Harmonic cohomology classes of symplectic manifolds, Comment. Math. Helv. 70 (1995), 1 - 9. [N] K. Nomizu : On the cohomology of compact homogeneous spaces of nilpotent Lie groups, Ann. of Math., 59 (1954), 531 - 538. [SY] Y. Sakane and T. Yamada : Harmonic forms on compact symplectic 2-step nilmanifolds, preprint. [TO] A. Tralle and J. Oprea : Symplectic Manifolds with no Kaher Structure, Lecture Notes in Mathematics 1661, Springer-Verlag, Berlin, 1997. [YM] T. Yamada : Harmonic cohomology groups on compact symplectic nilmanifolds, to appear in Osaka Math. J. [YN] D. Yan : Hodge structure on symplectic manifolds, Adv. in Math. 120 (1996), 143 - 154.
A SURVEY OF COMPLETE MANIFOLDS W I T H B O U N D E D RADIAL CURVATURE F U N C T I O N
KATSUHIRO SHIOHAMA
1. Introduction This is the survey Y.Machigashira and mannian n-manifold only if the metric is around p as
of my recent works jointly done by Y.Itokawa, M.Tanaka in [15], [16] and [29]. A complete RieM with a base point at p is said to be a model if and expressed in terms of the geodesic polar coordinates
otf = dt2 + f2(t)ds2sn-1(Q),
(t,Q) e (0,1) x S " - \ l
(1.1)
Here / : (0,£) ->• R is a positive smooth function satisfying /"(£) + K(t)f(t) = 0, /(0) = 0, /'(0) = 1 and ds|»-i is the standard metric on the unit (n — l)-sphere, and K : [0,£) —> R is the radial curvature function. Throughout this note let M be a complete Riemannian n-manifold with a base point p 6 M, at which the radial curvature is bounded below by K : [0,£) —> R. Namely, along every minimizing geodesic 7 : [0,/?) -» M with 7(0) = p, j3 < £ the sectional curvature KM satisfies KM(7'(t),X)
> K(t), W e [0,/3), VX e M T ( t ) , X l 7 ' ( i )
(1.2)
The notion of asymptotically non-negative curvature is a special case of our model. The topology of complete open manifolds with asymptotically non-negative curvature has been investigated by many people, see [14], [1], [2], [11], [18] and [27]. The topology of complete manifolds with radial curvature bounded below by a constant was first investigated by Klingenberg [19] and later by Machigashira [20], [21], [23] and recently by Zhu [36], Marenich and S.Mendonca [26] and others. Among them Abresch [1], [2] has established the angle comparison theorem at the corners x,y of A(pxy) C M of asymptotically nonnegative curvature. However the angle estimate at the base point is not perfect. The angle estimate at the base The work was partially supported by the Grant-in-Aid for General Scientific Research, No. 12440021 and for Exploratory Research, No. 13874021
216
217
point has been first proved by Machigashira [22] under weaker assumptions than asymptotically nonnegative curvature (see §2). Therefore the standard Toponogov comparison theorem holds in this case . The study of the poles of complete open surfaces of revolution of positive Gaussian curvature, which we call von Mangoldt surfaces was initiated by Hans von Mangoldt [25]. Then, Cohn-Vossen [9] proved that the set of poles of complete open surfaces of positive Gaussian curvature is compact. Cheeger, Gromoll and Meyer [13] and [6] then discussed the set of poles on complete open n-manifolds of positive curvature. Clearly a pole is a O-dimensional soul. It has been expected to estimate the size of the set of poles in terms of total curvature or other geometric meanings of complete open surfaces. Maeda [24] succeeded in giving the upper bound of the diameter of the set of poles on complete open manifolds of nonnegative curvature. Sugahara then gave sharper estimates in [31] and [32] including Alexandrov spaces with curvature bounded below by 0. The ideal boundaries of Hadamard manifolds have been constructed by Eberlein and O'Neil [10] by using the asymptotic relation of rays. As is seen in Busemann [4], the asymptotic relation of rays is not in general an equivalence relation. The Tits metric on the ideal boundary M(oo) of an Hadamard manifold M is given in [3] and [10]. Gromov [3] introduced a new metric on the ideal boundary Bd{M) of a complete noncompact manifold M by using the distance functions, where each element on Bd{M) is a Busemann function for a ray. Here the ideal boundary equipped with the Tits metric M(oo) of an Hadamard manifold M coincides with the Bd(M) consisting of the class of all Busemann functions on it. The ideal boundaries of complete open surfaces with total curvature has been investigated by Shioya [30]. We have established in [16] the Alexandrov-Toponogov comparison theorems for every geodesic triangle of the form A(pxy), where the corresponding triangle A(pxy) sketched on the model surface M2 is a generalized triangle. Such angle comparison theorem was obtained by Abresch [1] under stronger restriction of asymptotically nonnegative curvature. Other people followed his method. Further the standard Toponogov comparison theorem has been established when the model is a von Mangoldt surface of revolution, see Example 2.2. The maximal diameter theorem in our setting [15] states that M is isometric to a model Mn if the radial curvature of M at p is bounded below by K : [0,£) —> R and if supxeM d(p,x) = t < oo. Here K is the radial curvature function of the model M. This result has been extended to noncompact complete M by using the compactification M :— M U M(oo) of such an M. Here the cut locus C(p) to p has the
218
property C(p) = M(oo) and C(M(oo)) = {p}. It is interesting to investigate the relation between the total curvature, the integral condition (4.5), and the poles and the ideal boundary of complete open manifold with a base point p at which the radial curvature is bounded below by K : [0, oo) —> R. The basic tools in Riemannian geometry are referred to the books [4], [5], [3], [12], [14] and [28]. 2. Model surfaces Definitions and notations used in this note are prepared. Recall that a complete Riemannian n-manifold M with a base point at p € M is by definition a model if and only if its metric is expressed in terms of the geodesic polar coordinates around p as (1.1). If I < oo, then the compactness condition is required as follows:
m = o, /'w = - i
(2.i)
The compactness condition (2.1) implies that M is diffeomorphic to S", see [17]. In fact there is a unique point, say q G M furthest from p in such a way that the cut locus C(p) to p has the property that C(p) = {q},
C(q) = {p}
(2.2)
This property will suggest us the compactification of a noncompact model on which the maximal diameter theorem may hold. We emphasize that no restriction is imposed on our radial curvature function K. Our model surfaces may not admit total curvature. The K may change sign. We thus discuss much wider class of manifolds. Here is a natural question concerning with the equality on the radial curvature function (1.2). Let M be a complete n-manifold with a base point p. Let 7; : [0, /?») —> M for i = 1,2 be minimizing geodesies with 7J(0) = p and fa < oo. Assume that the sectional curvature of M satisfies #M(7i(*)>*0 = KM(t),X2),;Vt
G [0,ft), VX, e Mli(t),
Xt±^(t) (2.3)
Then, is M isometric to an n-model? The compactness assumption (2.1) imposes very strong restriction on the cut locus to p. Although the assumption (2.3) does not control the shape of C(p) but the metric, see [17]. Theorem 2.1. Under the assumption (2.3) M is either isometric to an n-model or else diffeomorphic to the real projective n space or a space of constant sectional curvature.
219 A unit speed geodesic 7 : [0,00) —> M is expressed by j(s) = (t(s),6(s)). Here the orientation of a geodesic is always taken so as to satisfy — P - > 0, for all s e [0,00) as The well known Clairaut relation for 7 is given by f(t(s))
• 6'(s) = C( 7 ) = f(t(s))\ s i n Z ( f (*), Vt(7(«)))|
(2.4)
where C(y) > 0 is a constant depending only on 7. Clearly (7(7) > 0 if and only if 7 does not pass through p. The equations of geodesic imply together with the Clairaut relation that
«
=
°™
>0
(2.5)
The Clairaut relation plays an important role in this note. 2.1. Examples
of models
Example 2.1. (Surfaces of revolution in R 3 ) Let c : [0,oo) —> R 2 be a unit speed smooth curve emanating from the origin and c(0) is tangent to the j/-axis and do not touch to z-axis except at the origin. Rotate it around x-axis to obtain a model surface immersed in R 3 . This M2 may not admit total curvature. There is no restriction on the sign of the radial curvature functions. Example 2.2. (von Mangoldt surface of revolution) A complete Riemannian 2-manifold M 2 is called a von Mangoldt surface of revolution iff it is a model surface whose radial curvature function is monotone nonincreasing on [0, t). The round sphere is the only compact smooth von Mangoldt surface of revolution. A compact von Mangoldt surface of revolution admits a unique singular point at the point q furthest from the base point. The tangent cone at q is generated by a circle of length —2nf'(£) < 2TT. Parabola and two sheeted hyperboloid are von Mangoldt surfaces of revolution. Example 2.3. We define a special von Mangoldt surface of revolution with total curvature 2-K on which every complete geodesic has at most finitely many self inter sections. Let T C R 3 be a C2-surface of revolution whose profile curve
J>2-l)/2
0
I rlogr
r > 1
220
Then setting r := yjx2 + y2, we observe that the radial curvature function K of T is: (TTZTW
0
< 1
Thus the total curvature satisfies c{!F) = 2-K and the length L(t) of the i-circle around the base point has the property lim^oo —£-<• = 0. 3. Generalized Alexandrov Toponogov Comparison Theorems We discuss the generalized triangle comparison theorems due to Alexandrov and Toponogov where the reference surfaces are models. The problem occurs when the corresponding triangle sketched on a model is not necessarily uniquely determined. Therefore we need to take up generalized geodesic triangles on models taking the form A(pxy) := Aijjxy) C M. By means of the Clairaut relation, we can always choose the most suitable such a generalized triangle. 3.1. Generalized
triangles
A geodesic triangle is by definition a triple of minimizing geodesic joining three points, called the corners of it. Two such points may be joined by many minimizing geodesies. We customary denote by A{xyz), without showing the edges but corners. This is because the angle comparison theorems hold independent of the choice of edge between two corners. We consider geodesic triangle of the form A(pxy) C M. Namely, one corner of every triangle is always chosen at the base point p of M. The corresponding triangle is always sketched on the model and expressed in two different manners. One is expressed as A(pxy) = A(pxy) and the other as A(pxy) = A(pxy). The corresponding triangle means that all the corresponding edges have the same lengths. Namely, if A(pxy) = A(pxy) C M is the corresponding triangle to A{pxy) C M, then d(p,x) = d(p,x), d{p,y) = d(p,y), \x,y\ = d(x,y)
(3.1)
Here \xy\ is the length of the edge xy of A(pxy) opposite p, and not necessarily minimizing. If A(pxy) — A(px ,y) C M corresponds to A(pxy) C M, then d(p,x) = d{p,x), d{p,y) = d(p,y), d{x,y) = d{x,y)
(3.2)
221
The triangles of the form A(p x y) = A(p x y) is employed when the model is a von Mangoldt surface of revolution. The triangles of the form A(pxy) — A(pxy) is employed when the model is not a von Mangoldt surface of revolution. The reason for this is explained by the appearance of the cut locus on the models. The cut locus to every point on a von Mangoldt surface of revolution has a simple structure, as stated (see [11] and [33]): Theorem 3.1. The cut locus C(x) to every point on a von Mangoldt surface of revolution has one of the the following properties : (1) C(x) = 0 (2) C(x) is a compact subarc of the meridian defined by 9 = 8(x) + n (3) C(x) is a subray of the meridian 0 = 9(x) + 7r 3.2. Alexandrov-Toponogov
Comparison
Theorems
We state the generalized Toponogov comparison theorems GTCT-I for general model surfaces and GTCT-II for von Mangoldt surfaces. Theorem 3.2. (GTCT-I) There exists for every geodesic triangle A(pxy) C M a generalized triangle A(pxy) = A(pxy) C M corresponding to A(pxy) such that Z{pxy)>
Z(pxy),
Z.(pyx) > l{pyx)
Here \xy\ is the length of the edge xy of
(3.3)
A(pxy).
The Alexandrov convexity theorem plays an essential role for the proof of GTCT-I. Let A(pxy) C M be an arbitrary fixed geodesic triangle and 7 : [0,1] —> M its edge such that 7(0) = x, 7(1) = y. If £ < 00, we then assume that the circumference of A(pxy) is less than 21. This will ensure the existence of the generalized triangle A(pxy) corresponding to A{pxy). With this notation we state the Theorem 3.3. (GACT) Let A(pxy) C M be an arbitrary fixed triangle (with its circumference L(A(pxy)) < 2£ if t < 00). Then there exists a l-parameter family {js : [0,1] -> M 2 } s£ [ 0]1 ] of geodesies such that 7.(0) = £,
t(x) = d(p,x),
t ( 7 , ( l ) ) = d(p,7(s)),
L(7.) = L(7|[0,s])
for every s £ [0,1] and such that the l-parameter family of generalized triangles {As := A(px7 s (l))} s e [ 0 ] 1 ] satisfies: (1) 7s[0,1] is the edge of As joining x to 7S(1)
222
(2) IfOx{s) := Z(-Vt(f),7g(0)), s E [0,1] is the angle at x ofAs, 0X is monotone non-increasing and lower semi continuous.
then
Remark 1. We observe from Theorem 3.3 that there exists a sequence of at most countable numbers 0 = so < Si < • • • < Sk < • • • < 1 such that 8x\[sie,Sk+i) is continuous monotone non-increasing and such that lim 9x(s)>ex(sk)
(3.4)
sfsfc
Here the discontinuity appears when the edge meets the cut locus to x. The edge is uniquely determined by the Clairaut relation (2.4). Remark 2. Let 7, 7 : [0,1] ->• M, M be the edge of A(pxy), A(px hy) opposite p, p respectively. We then observe the Alexandrov convexity property for distance functions: d(p>7(«))>*(7(*)).
Vae[0,l]
The following Lemma on narrow triangles is useful for the proof of the maximal diameter theorem in our situation. Lemma 3.1. (Lemma on narrow triangles) Let A(pxy) C M be a narrow triangle. Then there exists a unique triangle A{pxy) = A(pxy) C M corresponding to A{pxy) such that (3.3) is valid. Here one of the above equalities in (3.3) holds if and only if there exists a piece of totally geodesic surface in M bounded by A(pxy) and isometric to the interior of A(pxy). It should be remarked that the standard Toponogov comparison theorem holds when the model surface is taken to be a von Mangoldt surface of revolution or an Hadamard surface of revolution. This fact has first been discovered by Machigashira [22] for Hadamard case and developed in [16] for Mangoldt case, and summarized as follows. Theorem 3.4. (GTCT-II) Assume that the radial sectional curvature of M at p is bounded below by K : [0, €} —> R. Here K is the radial curvature function of a von Mangoldt surface of revolution or K < 0. Then there exists for every triangle A(pxy) C M a triangle A(pxy) = A(pxy) C M corresponding to A(pxy) such that t(x) = d(p,x),
t(y) = d(p,y),
d(x,y) = \xy\ = d{x,y)
(3.5)
Z(xpy)
(3.6)
and such that Z(pxy)>
/.{pxy),
Z{pyx)>Z(pyx),
> l(xpy)
223
In the extremal case where the equalities hold in each of the angle comparison on Theorems 3.2, 3.3 and 3.4, we have the following Corollary to Theorems 3.3, 3.2 and 3.4 Under the same assumption as in Theorems 3.3, 3.2 or 3-4 there exists a piece of totally geodesic surface bounded by A(pxy) which is isometric to the interior of the corresponding generalized triangle if and only if 6X is constant or one equality in (3.3) or (3.6) is satisfied. Remark 3. If the model is either a von Mangoldt or Hadamard surface of revolution, then we see from Theorem 3.1 that the angle function 6X : [0,1] ->• [0,7r] is continuous. This follows from the uniqueness of the corresponding geodesic triangle on the model. The uniqueness on a von Mangoldt surface of revolution follows from the fact that the generalized gradient vector fields to the distance function to x is transversal to Vt outside the cut locus to x, for details see [16]. 4. Maximal diameter theorem The Toponogov maximal diameter theorem states that a complete Riemannian n-manifold M with Ku > 1 is isometric to S" if its diameter satisfies diam(M) = ir. Also the Cheng maximal diameter theorem states that a complete n-manifold M is isometric to S" if RicM > n — 1 and if diam(M) = 7r. The maximal diameter theorem in our setting [15] is stated as follows. The proof needs the Lemma on narrow triangles as stated above. Theorem 4.1. M is isometric to an n-model Mn if the radial curvature of M at p is bounded below by K : [0,^) —> R and if I < oo and if sup d(p,x) — I < oo (4.1) xeM Remark 4. It follows from GTCT-I that there is a unique point q £ M such that sup d(p, x) = I xeM We then observe d(p,x)+d(x,q)=£,
VzeM
(4.2)
In particular C(p) = {q},
C(q) = {p}
(4.3)
224
The meaning of the relation (4.2) on the noncompact modd may be understood as follows. Suppose now that I -> oo. Then q £ M may be regarded as the limit point along an arbitrary fixed ray 7 : [0,00) -> M and the relation (4.2) will be understood as d(p,x) = lim [t - d(7(£),z)] =: F~(x) t—>oo
Thus the relation (4.2) on the noncompact model is regarded as the special one between the distance function to the base point and a Busemann function for 7. More precisely, if a noncompact M admits the compactification satisfying the maximal diameter theorem, then the above relation will be required for each ray. The meaning of the relation (4.3) on the noncompact model will be interpreted as follows. Clearly the first relation means that p is a pole and that the ideal boundary consists of a single point, say q. What is the cut locus to the ideal boundary? This is interpreted by the idea of Busemann (see : [4] ; §22). The set of all co-points to a ray 7 forms the singular set of Fy. Fortunately, the first observation states that all of the rays on M are asymptotic each other if and only if the ideal boundary consists of a single point. Theorem 4.2. A noncompact model surface M2 with base point at p admits the ideal boundary consisting of a single point, if one of the following conditions is satisfied: liminf f(t) = 0
(4.4)
t—>00 OO
/
r2{t)dt
= 00
(4.5)
Moreover we have C(M( 00)) = {p}
C(p) = M (00)
(4.6)
The conditions (4.4) and (4.5) are mutually independent. Stronger conclusion is derived from (4.5), as stated: Theorem 4.3. M is diffeomorphic to R™ if one of the conditions (4-4) or (4-5) is satisfied. Moreover M isometric to the n-model Mn if (4-5) is satisfied.
Remark 5. Under the assumptions in Theorem 4.3 we observe that both Bd(M) and M(oo) are single point.
225 References 1. U.Abresch, Lower curvature bounds, Toponogov's theorem and bounded topology, I, Ann. Sci. Ecole Norm. Sup. 19,(1985), 651-670 2. U.Abresch, Lower curvature bounds, Toponogov's theorem and bounded topology, II, Ann. Sci. Ecole Norm. Sup. 20,(1987), 475-502 3. W.Ballmann-M.Gromov-V.Schroder, Manifolds of nonpositive curvature, Progr. M a t h . 61(1985), Birkhauser, Boston-Basel-Stuttgart 4. H.Busemann, T h e Geometry of Geodesies, Academic Press, New York, 1955 5. J.Cheeger and D.Ebin, Comparison theorems in riemannian geometry, North-Holland Mathematical Library, 9,(1975) 6. J.Cheeger and D.Gromoll, On the structure of complete manifolds of nonnegative curvature, Ann. of Math. 96(1972), 413-443 7. S.Y.Cheng, Eigenvalue comparison theorem and its geometric application , Math.Z. 143(1975), 289-297 8. S.Cohn-Vossen, Kurzeste Wege und Totalkriimmung auf Flachen, Compositio Math. 2(1935), 63-133 9. S.Cohn-Vossen, Totalkrummumg und Geoddtische Linien und auf einfach zusammenhdngenden offenen volstandigen Fldchenstiicken, Recueil Math. Moskow (Mat. Sbornik),(1936), 139-163 10. P.Eberlein and B.O'Neil, Visibility manifolds, Pacific J. Math. 46(1973), 4 5 110 11. D.Elerath, An improved Toponogov comparison theorem for non-negatively curved manifolds, J.Differential Geometry, 15, (1980), 187-216 12. D.Gromoll-W.Klingenberg-W.Meyer, Riemannsche Geometrie im Grossen, Lecture Notes in Math. 55,(1968) Springer-Verlag, Berlin-Heidelberg-New York 13. D.Gromoll and W.Meyer, Complete open manifolds of positive curvature, Ann. of Math. 75(1969), 65-75 14. R.Greene and H.C.Wu, Function theory on manifolds which posesses a pole, Lecture Notes in Math. 69,(1979), Springer-Verlag, Berlin-Heidelberg-New York 15. Y.Itokawa-Y.Machigashira-K.Shiohama, Maximal diameter theorems for manifolds with restricted radial curvature, to appear in Proc. the Fifth R i m Pacific G e o m e t r y Conference, Sendai, (2001) 16. Y.Itokawa-Y.Machigashira-K.Shiohama, Generalized Toponogov's Theorem for manifolds with radial curvature bounded below, Preprint 2001 17. N.Katz and K.Kondo, Generalized space forms, to appear in Trans. Amer. Math. Soc. 18. A.Kasue, A compactification of manifolds with asymptotically nonnegative curvature, Ann.Scient. Ecole Norm. Sup. 21(1988), 593-622 19. W.Klingenberg, Manifolds with restricted conjugate locus, Ann. of Math . 78(1963), 527-547 20. Y.Machigashira, Manifolds with pinched radial curvature, Proc. Amer. Math. Soc. 118(1993), 979-985 21. Y.Machigashira, Complete open manifolds of non-negative radial curvature, Pacific J. Math. 165(1994), 153-160
226 22. Y.Machigashira, Manifolds of roughly non-negative radial curvature, preprint, (2001) 23. Y.Machigashira and K.Shiohama, Riemannian manifolds with positive radial curvature, Japanese J. Math. 19(1994), 419-430 24. M.Meda, Geodesic spheres and poles, Geometry of Manifolds, Perspectives in Math. 8(1989), 281-293, Academic Press, Boston 25. H. von Mangoldt, Uber diejenigen Punkte auf positiv gekrummten Flachen, welche die Eingenschaft haben, dass die von Ihnen ausgehenden geodatischen Linien nie aufhoren, kiirzeste Linien zu sein, J. Reine u. Angew. Math. 91(1881), 23-53 26. V.Marenich and S.Mendonca, Manifolds with minimal radial curvature bounded from below and big radius, Indiana Univ. Math. J. 48 (1999), 249-274 27. Y.Otsu, Topology of complete open manifolds with nonnegative Ricci curvature, Geometry of Manifolds, Perspecives in Math. 8(1989), Academic Press, Boston, 255-264 28. T.Sakai, Riemannian Geometry, Mathematical Monograph, Amer. Math. Soc. 8(1997) 29. K.Shiohama and M.Tanaka, Compactification and maximal diameter theorem for noncompact manifolds with radial curvature bounded below, to appear in Math. Z. 30. T.Shioya, The ideal boundaries of complete open surfaces, Tohoku Math. J. 43(1991), 37-59 31. K.Sugahara, On the poles of Riemannian manifolds of nonnegative curvature, Progress in Differential Geometry, Adv. Stud. Pure Math. 22,(1993), 321-332 32. K.Sugahara, On the poles of Riemannian manifolds and Alexandrov spaces of curvature bounded below, Kyushu J. of Math. 49(1995), 221-231 33. M.Tanaka, On the cut loci of a von Mangoldt's surface of revolution, J. Math. Soc. Japan 44(1992), 631-641 34. M.Tanaka, On a characterization of a surface of revolution with many poles, Memoirs Fac. Sci. Kyushu Univ. Ser. A, 46(1992), 251-268 35. V.A.Toponogov, Riemannian spaces having their curvature bounded below by a positive number, Amer. Math.. Transl. Ser. 37(1964), 291-336 36. S.H.Zhu, A volume comparison theorem for manifolds with asymptotically nonnegative curvature and its applications, Amer. J. Math. 112 (1994), 669682
Katsuhiro SHIOHAMA D e p a r t m e n t of Mathematics Faculty of Engineering Saga University Honjoh, Saga, 840-8525 J a p a n
YANG-MILLS C O N N E C T I O N S OVER COMPACT SYMPLECTIC MANIFOLDS
H A J I M E URAKAWA
Division of Mathematics, Graduate School of Information Sciences, Tohoku University, Aoba 09, Sendai, 980-8579, Japan, (e-mail: [email protected]) §0. Introduction In this note, we want to show the moduli space theory of Yang-Mills connections over compact symplectic manifolds. We first give a brif survey on Einstein-Hermitian connections on compact Kahler manifolds and strongly pseudoconvex CR manifolds. After giving materials of symplectic structures, we show characterization of anti-selfdual connections in terms of symplectic bivector field G in the case of four dimensional symplectic manifolds (cf. Propositin 5.1), and then consider the corresponding moduli space which included in the moduli space of YangMills connections over a compact symplectic manifold of any even dimension (cf. Theorem 5.3). Then, we determine the tangent space of our moduli space and show that the tangent space is of finite dimension (cf. Theorems 5.3, 5.8). Finally, we calculate explicitly the tangent space of the moduli space of flat connections over some compact solv-manifold which admits no Kaher structure (cf. Example 5.9). §1. Preliminary First, let us recall the Yang-Mills theory briefly. Let (E,h) be a vector bundle over a compact Riemannian manifold (M, g) with an inner product h. The Yang-Mills functional yM on the set C(E) of all connections of E is defined by
yM(V)
= l f \\RV\\\,
227
VeC(E).
228
A connection V G C(E) a Yang-Mills connection if it is a critical point of yM, i.e., for every one-parameter family V' G C{E), —e
-J dt
yM(V*) = 0. t=o
It is well-known for the four dimensional manifold, the special solution of Yang-Mills connection is known (cf. [D]). The star operator * of fl2(M) satisfies * 2 = Id, and it can be extended to the operator on the space fi2(F) of the endomorphism bundle F = End(E) valued 2-forms and fi2(F) can be decomposed into the eigenspaces with the eigenvalues ± 1 : n2(F) =
0,2+(F)®0.2_(F),
where ^ ( F ) is the eigenspace of * on fi2(F) with the eigenvalue ± , respectively. Let us recall Definition 1.1. A connection V is called self-dual (resp. anti-self-dual) if i ? v G n2+{F) (resp. R*(F) 6 il2_(F)), i.e., *fl v = Rv (resp. *RV = -JRV). Then it is clear that every self-dual or anti-self-dual connection is always a Yang-Mills connection. §2. Yang-Mills connections over Kahler manifolds For a compact Kahler manifold, let us recall the theory of EinsteinHermitian connections of a complex vector bundle. Let us take a holomorphic vector bundle E with the hermitian metric h over a compact Kahler manifold (M, g) of complex dimension n. Definition 2.1. A connection V € C{E) is a Hermitian connection of (E, h) if it satisfies the two following conditions: (i)
Xh(s,t)
= h(Vxs,t)
+ h(s,WYt),
X G T(T C M), s, t G T(E),
and (ii)
V ^ s = 8xs,
X G r(T°- 1 M), s e T{E),
where d is the holomorphic_structure of E, i.e., it satisfies the integrability f o r a11 condition: dY(dys) - 5 F ( % s ) = d^yf ^ Y e r(T°- 1 M) and lfi s G T(E). Here T M is the holomorphic tangent bundle and T°>lM is the anti-holomorphic tangent bundle of M, respectively, and X G r ( T 0 , 1 M ) is the complex conjugate of X G T{TlfiM).
229 Then, it turns out that the curvature tensor R? is of type (1,1), i.e., Rv e n M ( F ) , where QP'i(F) = T^T^M A A"T°'lM
A Hermitian connection V is Einstein if yf-ih.Rv v
for some constant c, or y/^lAR
=cld, = J27=i ^-V(ei^i)
G
^(F) is parallel
0
with respect to V, where (ei)x £ X^' M, and g(ei,ej) = <5„. It turns out that for a Hermitian connection V, it is a Yang-Mills connection if and only if it is Einstein. Furthermore, we have Theorem 2.3. (Kobayashi [Kb], Lubke [L], Donaldson [D], UhlenbeckYau [UY]) Let (E, h) be a holomorphic vector bundle with hermitian metric h over a compact Kahler manifold (M,g) of complex dimension n. Then, there exists an Einstein-Hermitian connection of (E, h) if and only if E is stable in the algebraic geometry, i.e., for all torsion free coherent subsheafS of E, ^ § < ££$., where deg(S) = JMCI(S) A *"~\ and $ is the Kahler form of (M, g).
§3. Yang-Mills connections over strongly pseudoconvex CR manifolds At the symposium in honor of Professor Su Buchin on his 90th birthday at 1991, Shanghai, we showed the odd-dimensional analogue of the EinsteinHermitian connections. Let us recall it, briefly (cf. [Url, Ur2, Ur3]). Let (M, £) be a compact CR manifold of real dimension 2n + 1, and 6 is a contact form on M. The complexification TCM is decomposed into TCM = S®S®C£, where SnS and [r(«S), r(<S)] C T(S). The contact form 9 satisfies that Sx 0 Sx = {X e T°M; 9(X) = 0}, x € M, and u = -dO is a non-degenerate on Sx ®SX, i.e., v = 9 Acon~1 is a volume element on M. Furthermore, we assume that (M, £) is strongly comvex, i.e., the Levi form L(X,Y) = u(IX,Y) is positive definite on Px = {X e TXM; 6{X) = 0}. We take a Riemannian metric on M by g(X,Y)=L(X,Y)+9(X)9(Y),
X,Y
&TXM.
A complex vector bundle E over M is holomorphic if there exists a differential operator d on E satisfying the integrability condition that d-x(dys) - dY(dYs)
- d[x,T\8 = °
230
for all X, Y G T(S) and s G T(£). For a Hermitian metric h on E, a connection V on £ is a Hermitian connection if it satisfies that V ^ s = djf-s for X g r ( 5 ) and s G r(.E), and X/i(s,t) = h(Vxs,t)
+ h(s,VTt),
I £ r ( T c M ) , a n d s,
teT(E).
Then we have (cf. [Url]) Theorem 3.1. Let (M, £) 6e a strongly pseudo-convex CR manifold, and E ne a holomorphic vector bundle with a Hermitian metric h on M. Let V be a Hermitian connection on (E, h) and the curvature tensor i ? v is of type (1,1), i.e., Rv G T(S* AS*
= w(X,Y),
ux{Xx,JXx)
> 0,
X,YeX{M), 0 ^ Xx e TXM,
where X(M) the space of all smooth vector fields on M. Define a compatible Riemannian metric g{X, Y) = u>(X,JY), X,Y £ X(M). For every point x G M, there exists a canonical coordinate (pi,--- ,pm,Qi,--- ,Qm) on a neighborhood U around x which satisfies m
to = ] P dpi A dqi and
(on
U),
231
By means of non-degeneracy of LJ, VM = ^r^" 1 is a volume element on M. We use the notion J a € fifc(M) for a £ fi fc (M), which is defined by {Ja)(X1,---,Xk)=a(JX1,---,Xk),
for Xi G X(M), (i = 1, • • • , k). Let G be the skew-symmetric bivector field on M given by jr[ dqi
dpi
and the symplectic star operator denoted by *s; ttk(M) -> Q,2m~k(M), defined in such a way that (aA*sp
= ~Ak(G)(a,p)vM,
is
a, 0 G Q,k{M),
y *s o *s = id Then, we can compare both the symplectic star operator * s and the usual star operator *. Lemma 4.2. For k = 2, we have *p = -*s
J{0 + {i{G)P)u),
0 G fi2(M),
where i{G)/3 is the interior derivation of (3 by G, for example, for /3 G
n\M), i{G)p = YZiKw<>4)
e
C
°°(M)-
§5. Yang-Mills connections over symplectic manifolds Now let (E, h) be a complex vector bundle with a Hermitian metric h over a compact symplectic manifold (M,w), and let C(E) be the space of all connections of E and C(E,h) = {V G C(E); Xh{s,t)
= h(Vxs,t)
+
h{s,WYt),
c
x e r(T (M)), s, t G r(E)}. For V G C(E,h), let i ? v be the curvature tensor field defined by RV(X,Y)s
= V x ( V y S ) - V y ( V x s ) - V [ J f ,y]«
for X, Y G 3E(M), and s G r ( £ ) . Note that B? G n
232
Proposition 5.1. Assume that dim(M) = 4. Then, for V G C(E,h), is anti-self-dual if and only if JRV = Rw
and
i{G)Rv
V
= 0.
Remark 5.2. We remark (cf. [I] and [D]) that in the Kahler surface M, and a complex vector bundle E, V is anti-self-dual if and only if JR7 = i ? v and Ai? v = 0. Our lemma is a natural generalization of the Kahler surface case. Thus, let us consider £{E,h) = {V eC(E,h);
JRV = Rv
and
i(G)Rv
= eld}
for some constant c. Note that £(E,h)
C {V G C(E,h); Yang-Mills connection}.
Let U(E,h) be the gauge group of (E,h) defined by U(E,h) = {
eC(E,h),
where Aut(E) is the automorphism group of the bundle E. Let us consider the moduli space £(E,h)/U(E,h), we denote by [V] the equivalence class containing V G £(E, h). Then, we have by a standard way, Theorem 5.3. For each V G £(E, h), the tangent space of the moduli space £(E,h)/U(E,h) at [V], denoted byT[v](£(E,h)/U(E,h)), is isomorphic to the space {a G Q1{F); 5va = 0, i(G){dva)
= 0, dwa e n 1 ' 1 ^ ) } ,
where <5V; n 1 ( F ) ->• T(F) is the formal adjoint of dv with respect to the global inner product induced from g, and Qp'g(F) is the space of F-valued forms on M of type (p, q) according to the decomposition ofT^M = T^'°M(B T®,lM, where T^'°M and T®'lM are the eigenspaces of the almost complex structure J with the eigenvalues \f—T and —•\/—T, respectively.
Since J is not integrable, in general, we should write, for each a = a' + a" <E n x ( F ) where a1 G nlfi(F) and a" G fi0-1^), as of a1 = d'va' + d'^a1 + dva',
d?a" = dva" + d'va" + d'^a" ,
233
according to the decomposition fi2(F) = ft2>°(F) © ft1-1^) © n°'2(F). also write as
We
for a = a' + a" £ O 1 ^ ) . We have for each a G fi^F), f dVQ' + ^V«" = 0
V7
and (5.5)
z(G)(d^a') + i{G){d'va")
= i{G)a = 0.
For the calculation of 6wa = 0, we have ^ a ' = -i(G)(d'^a') <^a" = i(G)(d'va")
+ Ja'(i(G)iV), -
±a"(i(G)N),
where N is the Nijenhuis tensor of J defined by N(X, Y) = [JX, JY] - [X, Y] - J[JX, Y] - J[X, JY],
X, Y <E X(M),
and m p. p. m i(G)JV = \^ Ni , i=l dqi dpi
But, since (4.1) J ( ^
= ( ^
and J ( ^
= - ( ^
, we have
z(G)./V = 0 at Z G M. So, we have <5Va' = - t ( G ) « a ' ) ,
<5>" = i(G)(d' v a") •
Thus, we have (5.6)
-i{G)(d'^a')
+ i(G)(d'va")
= S'va' + S'^a" = <5va = 0 .
The conditions (5.5) and (5.6) are equivalent to the following: i(G)(d£a') = 0,
hence we obtain
{6'va'
= 0,
234
Theorem 5.7. The tangent space of the moduli space T\y](£(E, h)/U(E, is isomorphic to the space la e ^(F);
h))
d'wa' + dwa" = 0,(1$ a" + dva' = 0,S'va' = 0,6$ a" = o j ,
which is also isomorphic to the space H 0 ' 1 ^ ) - j c / ' e f t 0 ' 1 ^ ) ; a$a" = ^tc7J oN,
S$a" = o j ,
where F — End(.E, h) is the bundle of skew-Hermitian endomorphism of (E,h) anda' = -tafT.
Furthermore, we have Theorem 5.8. The space H°'l(F)
is included in the space
| o " G fi0'1^); A$a" - \5$?^
o N) = o } ,
which is a finite dimensional space since A$ := d$S$ + S$d$ is still an elliptic differential operator acting on f20,1(.F). Finally, we give an example: Example 5.9. (cf [CFD]) Cordero, Fernandez and De Leon gave the following example of a compact symplectic manifold which admits no Kahler structure. Let G := H(l,r) x S1 be a closed connected solvable Lie subgroup of GL(r + 3, C) which is defined by ( 1
G= I A = 0 • • \o • •
0 0
ox
6i
0
ar 1 0 0
br br+i 1 0
0 0 0 e2n
\ ; a,i,bi,b e R > ,
7
and T is a discrete subgroup of G defined by T = {AG G; a,i,bi, a e Z} , and the quotient space M — G/T is a compact manifolds. We denote the natural projection from G onto M by it. Let us denote by x%, y, z% and t (i = 1, • • • , r), the coordinate functions on G defined by Xi(A) - ait y{A) = br+l, Zi(A) = bit t(A) = a
235
for every A € G (i = 1, • • • ,r). The 1-forms 5;, /?, 7^ and 77 on G defined by 5; = dxi, P = dy , 7J = dz, - Zioh/, rj = dt are left invariant. Therefore, there exist (2r + 2) 1-forms ctj, /? ,7* and 77 on M satisfying that lT*ai = OLi , 7T*/? = P , 7T*7i = 7 j , 7T*77 = 7j.
The 2-form on M defined by r
a; = /?A77 + y"^a»A7i i=l
gives a symplectic structure on M. We denote by Xi, Y, Z{ and T, the dual fields on M of { a i , /3, 7;, 77}. The almost complex structure defined by is compatible to w. We take a trivial bundle E = M x C s . Then, H0tl(F) is isomorphic to the space Hl(M) ® {C e M(s, C); C+ tC = O}, and d i m a f f 0 ' 1 ^ ) = dimR, i I 1 ( M ) • s 2 = (r + 2) • s 2 , i.e., the dimension of the moduli space, dim(T[ V ](£(E, h)/U(E, h)) is (r + 2) • s 2 , where ff^M) is the first de Rham cohomology of M.
REFERENCES [B]
J-L. Brylinski, A differential complex for Poisson manifolds, J. Differ. Geom. 28 (1988), 93-114. [BDU] E. Barletta, S. Dragomir and H. Urakawa, Pseudoharmonic maps from nondegenerate CR manifolds to Riemannian manifolds, Indiana Univ. Math. J. (2002) (to appear). [CFD] L.A. Cordero, M. Fernandez and M. De Leon, Examples of compact non-Kdhler almost Kdhler manifolds, Proc. Amer. Math. Soc. 95 (1985), 280-286. [D] S.K. Donaldson, Anti-self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles, Proc. London Math. Soc. 50 (1985), 1-26. [I] M. Itoh, On the moduli space of anti-self-dual Yang-Mills connections on Kdhler surfaces, Publ. RIMS, Kyoto Univ. 19 (1983), 15-32. [Kb] S. Kobayashi, Differential Geometry of Complex Vector Bundles, Iwanami, Tokyo, 1987. [Ks] J-L. Koszul, Crochet de Schouten-Nijenhuis et cohomology, Asterisque (1985), 257-271. [L] M. Lubke, Stability of Einstein-Hermitian vector bundles, Manusc. Math. 42 (1983), 245-257.
236 [M] [UY]
[Url] [Ur2]
[Ur3]
[Y]
O. Mathieu, Harmonic cohomology classes of symplectic manifolds, Comment. Math. Helv. 7 0 (1995), 1-9. K. Uhlenbeck and S.T. Yau, On the existence of hermitian Yang-Mills connections in stable vector bundles, Commun Pure Appl. Math. 39 (1986), S257S293. H. Urakawa, Yang-Mills connections over compact strongly pseudoconvex CR manifolds, Math. Z. 216 (1994), 541-573. , Variational problems over strongly pseudoconvex CR manifolds, Differential Geometry, Proceedings of the Symposium in Honor of Professor Su Buchin on his 90th Birthday (C. H. Gu, H. S. Hu and Y. L. Xin, eds.), World Scientific, Singapore, 1993, pp. 233-242. , Yang-Mills connections and deformation theory over compact strongly pseudoconvex CR manifolds, Geometric Complex Analysis, Third Intern. Res. Inst. Math. Soc. Japan (J. Noguchi et al., eds.), World Scientific, Singanore, 1996, pp. 635-652. D. Yan, Hodge structure on symplectic manifolds, Advances Math. 120 (1996), 143-154.
NONLINEAR SCHRODINGER SYSTEMS ASSOCIATED W I T H H E R M I T I A N S Y M M E T R I C LIE A L G E B R A S
HONG-YU WANG Department of Mathematics, Yangzhou University Yangzhou, Jiangsu 225002, People's Republic of China E-mail: [email protected] In this article, we discuss vortex filaments, Heisenberg models and nonlinear Schrodinger equations. We also describe the correspondence among these three integrable systems with (quasi)-periodic boundary condition.
1. T h r e e classical i n t e g r a b l e s y s t e m s In this section, we briefly recall the three classical integrable systems which have many applications in physics, have been widely studied by m a t h e m a t i cians and physicists, see e.g. [1, 10, 19].
1.1. Heisenberg
model
In physics, the ferromagnetic or Heisenberg spin chain system is modeled by du — =ux
Agu,
. . u[-,0) = u0,
. . (1)
where un is a smooth m a p from Riemannian manifold (N, g) into the unit sphere S 2 in R 3 , " x " is t h e cross product for vectors in R 3 and Ag is the Laplacian operator on N induced by Riemannian metric g. If N = R 1 or S1, Equation (1) is an integrable system [2, 10].
1.2.
Vector filament equation
and cubic
nonlinear
Schrodinger
Another example arose in the study of vortex filament in the work of D a Rios (1906, see [19]). Consider the evolution of curves in R 3 according to the vortex filament equation at = ax x axx.
237
(2)
238
It is easy to see that la^l 2 is preserved under the evolution. It follows that a(-, 0) is parametrized by its arc length, then so are all a(-, t) to all t. Hence, it evolves in time along its binormal according to the equation ^
=
k(a)B(a),
where k(a) is the curvature and B(a) is the binormal of a. Differentiating the vortex filament equation (2) with respect to the arc length parameter x, we obtain the following equation for the unit tangent vector T:
Tt=TxTxx. This equation is known as the Heisenberg model (1). Of particular interest to us is its relation with the nonlinear Schrodinger equation below: qt = i(qxx + 2\q\2q),
(3)
where q(x,t) is a complex-valued function defined by q(x,t)=k{a)exp(
-r(a)),
(4)
and where r(a) is the torsion of a. Equation (4) is called Hasimoto transformation (1972, cf. [14]). 2. Schrodinger-like systems associated with Hermitian symmetric Lie algebras 2.1. Hermitian
symmetric
spaces
We first recall some important facts about Hermitian symmetric spaces (For details, see [13].). A homogeneous space of a Lie group G is a differentiable manifold M on which G acts transitively. The subgroup of G which leaves a given point po £ M fixed is called the isotropy group at po. Denoting the isotropy group by K, then M S G/K. At the level of Lie algebras, let g and 6 be the Lie algebras of G and K respectively, and let m be the vector space complement of 6 in g. Then
g = m 9 ! with [E,t] C t, and m is identified with the tangent space TPoM at the point po. If furthermore, (g,6) satisfies the conditions [4,m] € m and [tn, m] C fi, then g is called a symmetric Lie algebra and G/K is a symmetric space. For a symmetric space, a metric is given by killing form g(X,Y)
=
tr(adXadY),
239 and, at the fixed point po, the corresponding Riemannian curvature tensor is given by (R(X,Y)Z)P0
= -[[X,Y},Z],
X,Y,Ze
m,
where we note that [X, Y] 6 6. We are particularly interested in these symmetric Lie algebras which have a complex structure. This is a linear endomorphism J : m -> m satisfying J 2 = —id. Within this class, one has the corresponding Hermitian symmetric spaces. These spaces have the property that there exists a special element a G t such that t — Cg(a) := {£ 6 0 : [£,a] = 0 } and such that J = ad a\m. 2.2. Heisenberg symmetric
models with values in spaces
Hermitian
Now, let M = G/K be a compact (irreducible) Hermitian symmetric space with complex structure J = ad a\m at po- Then, M has a standard isometric embedding as an Adjoint G-orbit space at a in g M = G/K^
{gag"1
:9eG}.
Hence, at u = gag"1 € M, the complex structure on TUM is given by J(u) — ad u. Let C ( R 1 , M ) denote the space of smooth path u : R 1 -» M where ux is in the Schwartz class 5 ( R 1 , T M ) . Then the generalized Heisenberg model with values in the symmetric space M (Y. Nakamura [17], 1985) can be expressed as the question below (cf. [16, 22, 23]). ut = [u,uxx\. 2.3. Sym-Pohlmeyer
(5)
curves
The generalized vortex filament equation with values in the compact Hermitian symmetric Lie algebra g is given by lt = bfx,7xx]-
(6)
A special case is that the x-derivative j x lies in the Hermitian symmetric space G/K, then -yx satisfies Equation (5) (for the generalized Heisenberg model with values in compact Hermitian symmetric space M = G/K). This kind of curve is called Sym-Pohlmeyer and becomes an arc length parameter if the killing form of g is properly rescaled (cf. [16]). For curves in R 3 (= Lie algebra of SU(2)), this is true if the curves are parametrized by arc length (Note that S2 =* SU(2)/U(1).).
240
2.4. Nonlinear Hermitian
Schrodinger equation associated symmetric Lie algebra
with
The analog of nonlinear Schrodinger equation associated with Hermitian symmetric Lie algebra g is first induced by Fordy-Kulish [11]. Let g — m©£ be the Cartan decomposition of the compact Hermitian symmetric Lie algebra g, and let a be the special element. Then the generalized nonlinear Schrodinger equation associated with g is given by Vt = Jr)xx + ^[v,[TV,v]],
(7)
where r)(x,t) g m (called m-potential), and J = ada\m is the complex structure on m.
3. Gauge equivalence 3.1.
Example
For G = U(n), choose _ ( \ h
0
\
""ID -i/„-JThen the Hermitian symmetric Lie algebra is: u(n) = u t t ( n ) © u ^ n ) - 1 , where u»(n) = {f € u ( n ) : K , a ] = 0}, and Ua(n)X = {V € u(n) : fa,0 = 0,V^ G u a (n)}
= {(-°^o) :BeM M-)}The corresponding compact Hermitian symmetric space is the Grassmannian Gr{k, C n ) and Equation (7) becomes the cubic matrix nonlinear Schrodinger equation [22, 23] Bt = i(Bxx + 2BB*B).
(8)
241
3.2.
Hamiltonians
Given a solution, u, of Equation (5), as the domain R 1 x R 1 is simply connected, we can choose a smooth map g(-,t) : R 1 —> G such that u — gag~l. Note that G/K can be regarded as an Adjoint G-orbit space at a in the Lie algebra g, g(—oo,t) = e (the unit element), and g~xgx £ ni. Set r] = g~1gx, then rj is a solution of Equation (7) by a direct computation [23]. Note also that the second Hamiltonian of u is defined by
H{u) = /
{-\[u,uxx]\2
+-R(ux,J(u)ux,ux,J(u)ux}
= /
{-\Vx\2 + ~:R(V, Jr),ri,Jv)po} 4
= /
{~\rix\2 +
JR 1 JR.1
4
\\[v,Jri}\2}
:= H(V).
Here H{rj) is the Hamiltonian of rj. Hence Equation (7) is a Hamiltonian equation with the Hamiltonian H{rj) (cf [23]). We would like to point out that as a conservation law for H(u) can be extended to the target space that is a Hermitian locally symmetric space [18].
3.3. Gauge
equivalence
From the discussion in the above, it is easy to see the gauge equivalence between Equations (5) and (7) [22, 26]: Proposition 1 The Heisenberg model with values in compact Hermitian symmetric space G/K (see Equation (5)) is gauge equivalent to the generalized nonlinear Schrodinger equation on the real line R 1 associated with the compact Hermitian symmetric Lie algebra g (see Equation (7)). rWith Proposition 1 understood, we may think that the Heisenberg model (Equation (5)) is to be regarded as the intermediate model between Sym-Pohlmeyer equation (Equation (6)) and the nonlinear Schrodinger equation (Equation (7)) [16]. Hence, to study the relation among Equations (5)-(7), we should first consider the correspondence between Equations (5) and (7).
242
4. The correspondence between Heisenberg model and nonlinear Schrodinger equation with (quasi)-periodic boundary condition This section is devoted to discussing the correspondence among given three integrable systems. For simplicity, let gVF, gHM and gNS denote SymPohlmeyer curve, Heisenberg model with values in Hermitian symmetric space and nonlinear Schrodinger equation associated with Hermitian symmetric Lie algebra. As done in the classical case [2, 12], it is natural to ask the following question. Question Which kind of solutions of gNS can be realized as closed curves evolving by gVF? 4.1.
Notation
Let S,** denote the solution space, for example, SgHM = {solutions of gHM}. Moreover, S
°gHM = { « € SgHM • U{X + 27T, t) = u(x, t)}.
Without loss of generality, we always assume that the period is 27r. It is easy to see: /•27T
S
°VF "^> {M £ S°gHM : / Jo
u(x,t)dx
= 0},
but we do not know that S°VF —>• which subset of S9NS- Here 7i denotes generalized Hasimoto transformation which will be defined shortly. To solve the problem in the above, the basic idea is as follows: Regard gHM as intermediate model of gVF and gNS, first find S°HM
—> which subset of SgNs?
Here Q is a suitable gauge transformation since gHM is gauge equivalent to gNS. It turns out that the relevant subset should satisfy a compatibility condition [4], that is, SgNS = {77 S SgNS '• V is fc-quasi-periodic and fc-compatible for some constant k G K). Definition 1 rj(r, t) is said to be /c-quasi-periodic for some constant k € K if the following equality holds: •q{x + 2TT, t) = k~lr](x, t)k.
243
Furthermore, TJ(X, t) is also fc-compatible if the following system of h(x) 6 G is solvable: (h-1hx = rj(x,0), \ h(0) = e(unit element), h(2?r) = k. 4.2.
The correspondence
among gVF,
gHM
and
{
. '
gNS
With the compatibility condition understood, we now discuss the correspondence among gVF, gHM and gNS [4]. Theorem 2 Under the gauge equivalence between gHM and gNS, there is a 1-1 correspondence between S0HM/Ad{G) and ScNS/Ad(K). By means of Theorem 2, we have a geometric realization of gNS as closed Sym-Pohlmeye curves: Theorem 3 A solution r\ of gNS corresponds to a periodic solution of gVF with period 2ir (under % _ 1 ) if and only if the following two conditions hold: (1) W € SgNS> i-e-' *7 ' s ^-quasi-periodic and ^-compatible for some constant k £ K. (2) | 0 *«(a;,0)(ii = 0, where u(x,t) is a representative of
is a conserved quantity: u(x,t)dx
= / J0
f
ut(x,t)dx
C2TT
/o
Jo/•27T
[u,uxx]dx £
= [u(z,i),u a; (a;,t)]o' r = 0. Since u is periodic (in spatial variable x) with period 2TT by Theorem 2. (2) With Theorems 2-3 at hand, it is easy to see that % can be written in the form H = Q odx. 4.3.
Example
Consider the geometric realization of the (classical) cubic nonlinear Schrodinger equation closed vortex filament curve in R 3 . The Hermitian
244
symmetric space 5 2 = G/K will be G = 5(7(2) and
Hence the Cartan decomposition of Hermitian symmetric Lie algebra is su(2) = m © t, where
and
-{(•I;)=.£c}aa The complex structure on m is given by J = ada\m, where (**
0
\
Then the NS, qt = i(qXx + 2|
The HM, J-t
=Z
-*• X ^
J-XXl
is equivalent to gHM, ut = [u,uxx], with 1 /
itz
*i + it2 u
2 V—*i + 2
-ih
where
T= | t2 I e S 2 c R 3 . ,*3.
J'
245
The VF, Q-t — &x
*
OLXX,
is equivalent to gVF, ^t — [7"x j Txx\i
with _ 1 / ia3 a i + ia2 2 \ - a i + ia2 -iA 3 where a =
G R3.
a2
In summary of the above discussion, we have: Corollary 4 Under the Hasimoto transformation from VF to NS, a solution q of TVS corresponds to a closed curve with length 2TT evolving by VF if and only if the following the conditions hold: (1) rj = I
J is /c-quasi-periodic and fc-compatible for some constant
(2) JQ* u(x,0)dx
is a representative of t ? - 1 ^ ) £
= 0, where u(x,t)
S» ffM /Ad(Stf(2)). Remark In fact, we can adopt the method (see [2, 12]) that is to use natural frame instead of Serret-Frenet frame of space curve to prove the above equivalence. 5. Concluding remarks 5.1. Outline of proof of Theorem
2
To prove Theorem 2, we should check [4]: SgHM **
SgNs.
(1) "-»": given u £ S°gHM, solve (g,rj) where g{x,t) e G, rj(x,t) £ m of the following system: ugag-1, x 9~ gx ~ V € m, Q~l9t = Jr)x + h[Jri,rj\-
(10)
246
Here J = ada\m. Claim:
u £ S°HM
=> r) £
S°NS.
(2) "<—": given r? € SgNS, first solve g(x,t) 6 G of the following system: S " 1 ^ = f,
( n )
l
9 9t = JVx + \{Jr],rj\. Then, define u — gag-1. Claim:
ScgNS £ u £ S°gHM
=* IJ.
Remark Note that the systems (10) and (11) are overdetermined systems, hence they are solvable because of fc-compatibility condition [4]. 5.2.
Some
questions
With Theorems 2-3 and Corollary 4 at hand, the following questions are of particular interest to us: (1) Backlund and Darbouk transformations with (quasi)-periodic boundary condition (to find new solutions form old ones, cf [2, 21]) (2) Algebro-gemetric method (to construct explicit (quasi)-periodic solutions in terms of Riemann theta functions, see [2, 5, 9, 12, 15, 25]) (3) Inverse scattering transformation and Poisson ations (see [10, 20]) (4) Whether every completely integrable system admits a biHamiltonian structure (in particular, periodic gHM, cf. [2, 10]) (5) Comparison between the gHM and gNS phase spaces (that is, comparison between S°gHM/Ad{G) and ScgNS/Ad(K), cf. [2, 22]) 5.3.
Heisenberg
model in higher
dimensions
Roughly speaking, Heisenberg model can be regarded as a Schrodinger-like system, hence we can define Schrodinger flow as follows. Let (N, g) be a Riemannian manifold, and let (M,w,J) be a Kahler manifold. Given a map u0 : N -> M, the Schrodinger flow of u0 is defined by the Cauchy problem [3, 6, 7, 18, 24]: -^
= J(U)T(U),
U(;0)
= UO,
(12)
where r(u) is the tension field of u; in local coordinates,
* . ) = A f „' + , - < r j , | £ | £ .
(13)
247
We note the analogy of (12) with the heat flow for harmonic maps. We also remark that the Schrodinger flow can be formulated more generally on symplectic manifolds [6]. If M is a Hermitian locally symmetric space, and N is S1 or R 1 . Then, given a map u0 from TV into M, the Cauchy problem for Schrodinger flow u(.,t) (see Eq. (12)) admits a unique global solution (see [6, 18, 24]). We may also consider the extension of the Schrodinger flow to higher dimensions. We, in fact, establish a local existence theory for the Cauchy problem of the Schrodinger flow from any compact Riemannian manifold into a Kahler manifold [7]. Finally, we would like to point out that if the target space is a compact Hermitian symmetric space (that can be regarded as an Adjoint G-orbit space in Lie algebra Q) , the equation for Schrodinger flow from a Riemannian manifold can be expressed as ut —
J(U)T(U)
=
[U,T(U)}
— [u,Au],
(14)
This is the generalized Heisenberg model with values in Hermitian symmetric space [23]. With Equation (14) at hand, we can prove the global existence of weak solutions to Schrodinger flows for maps from compact Riemann surfaces into compact Hermitian symmetric spaces [8]. Hence, the uniqueness of the weak solutions is also of particular interest to us. Acknowledgments The author wish to thank Professor Gu Chaohao and Professor Hu Hesheng for their encouragement. References [1] V. I. Arnol'd and B. A. Khesin, Topological Methods in Hydrodynamics, Appl. Math. Ser. 125, Springer-Verlag, Berlin-Heidelberg-New York, 1998. [2] A. Calini, Recent developments in integrable curve dynamics, in Geometric Approaches to Differential Equations (P. J. Vassilion and I. G. Lisk, eds), pp 56-99, Austral. Math. Soc. Lecture Ser. 15, Cambridge University Press, 2000. [3] N. Chang, J. Shatah and K. Uhlenbeck, Schrodinger maps, Comm. Pure Appl. Math. 53 (2000), 590-602. [4] B. Dai, H.Y. Wang and Y.D. Wang, The vortex filaments, Schrodinger flows and nonlinear Schrodinger equations associated with Lie algebra, preprint.
248
[5] B. Dai, H.Y. Wang and Y.D. Wang, An algebra-geometric approach to generalized Heisenberg models, in prepraration. [6] W. Y. Ding and Y. D. Wang, Schrodinger flows of maps into symplectic manifolds, Science in China A 41(1998), 746-755. [7] W. Y. Ding and Y. D. Wang, Local Schrodinger flow into Kahler manifolds to appear in Science in China A. [8] W. Y. Ding, H. Y. Wang and Y. D. Wang, Global weak solutions to Schrodinger flows on compact Hermitian symetric spaces, in prepraration. [9] B. A. Dubrovin, Theta functions and nonlinear equations, Russ. Math. Surv. 36 (1981), 11-92. [10] L. Faddeev and L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons, Springer-Verlag, Berlin-Heidelberg-New York, 1987. [11] A. P. Fordy and P. P. Kulish, Nonlinear Schrodinger equations and simple Lie algebras, Comm. Math. Phys. 89(1983), 427-443. [12] P. G. Grinevich and M. V. Schmidt, Closed curves in R 3 : a characterization in terms of curvature and torsion, the Hasimoto map and periodic solutions of the filament equation, preprint (dg-ga/ 9703020). [13] S. Helgason, Differential Geometry and Symmetric Spaces, Academic Press, New York, 1978. [14] R. Hasimoto, A soliton on vortex filament, J. Fluid Mech. 51 (1972), 477-485. [15] I. M. Krichever, Methods of algebraic geometry in the theory of nonlinear equations, Russ. Math. Surv. 32 (1977), 185-213. [16] J. Langer and R. Perline, Geometric realizations of Fordy-Kulish nonlinear Schrodinger systems, Pacific J. Math. 195 (2000), 157-178. [17] Y. Nakamura, On Heisenberg models with values in Hermitian symmetric spaces, Lett. Math. Phys. 9(1985), 107-112. [18] P. Y. H. Pang, H. Y. Wang and Y. D. Wang, Schrodinger flow on Hermitian locally symmetric spaces, to appear in Comm. Anal. Geom. [19] R. L. Ricca, Rediscovery of Da Rios equations, Nature 352(1991), 561562. [20] C. L. Terng and K. Uhlenbeck, Poisson action and scattering theory for integrable systems, Surveys in Differential Geometry: Integral Systems [Integrable Systems], pp 315-402, International Press, Boston, 1998. [21] C. L. Terng and K. Uhlenbeck, Backlund transformations and loop group actions, Comm. Pure Appl. Math. 53(2000), 1-75. [22] C. L. Terng and K. Uhlenbeck, Schrodinger flows on Grassmannians, preprint (DG/ 9901086). [23] H. Y. Wang, Geometric nonlinear Schrodinger equations, to appear Proceedings of the 9-th MSJ-IRI III.
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[24] H. Y. Wang and Y. D. Wang, Nonautonomous Schrodinger flows on Hermitian locally symmetric spaces, to appear in Science in China A. [25] M. A. Wisse, Quasiperiodic solutions for matrix nonlinear Schrodinger equations, J. Math. Phys. 33 (1992), 3694-3699. [26] V. E. Zakharov and L. A. Takhtajan, Equivalence of a nonlinear Schrodinger equation and a Heisenberg ferromagnetic equation, Theore. Math. Phys. 38(1979), 17-23.
O N T H E HENSEL LIFT OF A POLYNOMIAL
ZHE-XIAN WAN ACADEMY OF MATHEMATICS AND SYSTEM SCIENCES, CHINESE ACADEMY OF SCIENCES, BEIJING 100080, CHINA E-MAIL: [email protected]
Abstract Denote by R the Galois ring of characteristic pe and cardinality pem, where p is a prime and e and m are positive integers. Let g(x) be a monic polynomial over Fpm. A polynomial f(x) over R is defined to be a Hensel lift of g(x) in R[x] if f(x) = g(x), where — is the natural homomorphism from R onto Fp™, and there is a positive integer n not divisible by p such that f(x) divides xn — 1 in R[x]. It is proved that g(x) has a unique Hensel lift in R[x] if and only if g{x) has no multiple roots and x / g(x). An algorithm to compute the Hensel lift is also given. Index terms Finite field, Galois ring, Hensel lift of a polynomial 1. Introduction We begin with a review of the Galois Rings. Let Zpe = Z/(pe), where p is a prime and e is a positive integer. Denote by [p) the principal ideal of Zp« generated by p, then Z p e/(p) ~ F p . We have the natural homomorphism, denoted by —, - : Zp« -> F p a i-)- a = a + (p) and the extended homomorphism, denoted also by —, -:Zp.[x]->Fp[i] / = a 0 + a i z -I
n
1- anx
_ H> / = a 0 + ai£ -I
h a„a; n ,
where ao, ai, • • • , an G Z p e. Let h G Zp« [i] be a monic polynomial of degree m > 0 and assume that h is irreducible over F p . Then Zp«[x]/(/i) is the Galois ring of characteristic pe and cardinality pem. Let £ = x + (h), then Zp.[x]/(h) = Zpe[^]. Galois rings were first studied by W. Krull[2] in 1924. The following are special cases of Galois rings, (a) When e = l,Zp[(\= Fp[£] = Fp™.
250
251
(b) When m = 1, Zpe[£] = Z p e. In particular, when p = 2 and e = 2, we get Z4. In 1995 the following definition of the Hensel lift of a polynomial appeared in [1]. Let hi £ ¥2 [x] be of degree m > 0 and assume that /121 (%l - 1) and I is minimal subject to this property. There is a unique monic polynomial h € ln[x] of degree m such that h = ft2 and h\(xl - 1) in Z^x]. This polynomial is called the Hensel lift of /i 2 . In the above definition the condition that / is odd should be added. A counter-example when I is even is /i 2 = (x — l ) 2 (a;2 + x +1) with /121 (x6 — 1) in F 2 [x]. Now both h = {x2 - l){x2 + x + 1) and h' = {x2 - l)(x2 - x + 1) are in Z^[x] such that /i = h' = /i 2 and /i|(a;6 - 1), /i'|(a;6 - 1) in ln[x\. In the following we will suggest a simpler definition which can be formulated for an arbitrary Galois ring. ^From now on denote the Galois ring Zpe [£] by R. The homomorphism (1) can be extended to the natural homomorphism - : i? = Z p «[£] ^ F p ~ = F p £ ] ao + ai^ H
1- a m - i £
m_1
_ >-> 00 + ai^ H
_ ^ h a m _i£
,
where ao,ai, • • • , a m _ i € Zp«, which can then be extended to the homomorphism — : R[x] —>• Fpm[x] a0 + aix + • • • + anxn
... 1-4 a 0 + ~a.\x + • • • +
n
anx ,
where cto,ai,--- ,an € R. Then we formulate the following definition. Let g(x) be a monic polynomial over Fpm. A monic polynomial f(x) over R is called a Hensel lift of g(x) if f(x) = g(x) and there is a positive integer n not divisible by p such that f(x)\(xn — 1) in R[x]. In Section 2, it is proved that g(x) has a unique Hensel lift if and only if g(x) has no multiple roots and a; / g(x). An algorithm to compute the Hensel lift is given in Section 3. 2. Existence and Uniqueness Proposition 1. A monic polynomial g(x) over Fpm has a Hensel lift f(x) over R if and only if g(x) has no multiple roots and x J( g(x) in Fpm[a;]. Proof. Let f(x) be a Hensel lift of g(x) over R. Then f(x) = g(x) and there is a positive integer n not divisible by p such that f(x)\(xn - 1) in R[x]. From /(a;)|(a: n - 1) in R[x] we deduced 7{x)\(xn - 1) in Fpm[a;], i.e., g(x)\(xn — 1) in Fpm[a;]. Since p J[ n, xn — 1 has no multiple rooots and
252
g(x) has also no multiple roots. Since x J( (xn — 1) in Fpm [a;], we have also x I g(x) in Fpm[a;]. Conversely assume that g(x) has no multiple root and that x /\ g{x) in Fpm [a;]. Then there is a positive integer n not divisible by p such that g(x)\(xn - 1) in Fpr»[a:], (see [5], Definition 7.2 and Theorem 7.8). Assume that over Fp™ the polynomial xn — 1 is factored into xn - 1 = gi(x)g2(x) •••gr(x),
(5)
where g\{x),g2{x), • • • ,gr(x) are monic irreducible polynomials over Fpm and are uniquely determined by xn - 1 up to a rearrangement. Since p / n, xn — 1 has no multiple roots. It follows that g\(x), g2(x), • • • ,gr(x) are distinct. By Hensel's Lemma, there are monic polynomials fi(x)J2(x), ••• ,fr(x) £ R[x] such that xn-l
= fi(x)f2(x)
• • • fr(x) in R[x],
fi(x) =9i{x),i
(6)
= 1,2,-•• ,r,
(7)
and A ix), f2(x), • • • , fr(x)
are pairwise coprime in R[x].
n
Since g(x) \ (x — 1) in Fpm [x], by the unique factorization theorem in Fp™ [x], we can assume that g{x)= g\(x)g2{x)---gs(x),
where 0 < s < r,
up to a rearrangement of gi{x),g2{x), • • • ,gr(x). 9i(x)g2(x) • •• gs(x) = 1, when s = 0. Let
(8)
We agree that
f(x) = h(x)f2(x)---fs(x).
(9) n
then f(x) is a monic polynomial over R dividing x - 1 in R[x] and f(x) = g(x). Therefore f(x) is a Hensel lift oi g(x). L e m m a 2. Let ri\ and n2 be positive integers and n = gcd(ni, n2)• Then xn-l= gcd(a;"1 - l,xn* - 1) in Wpm[x], (xn - l)\{xni - 1) in R[x], and (a;"-1)|(a;" 2 - 1 ) in R[x]. Proof. We apply induction on max{ni,ri2} . When n\ = n2 , our lemma holds obviously. Let ni > n2, then max{ni - n2,U2} < max{ni,n 2 } = n\ and gcd(ni - n2,n2) = gcd.(ni,n2) = n. By induction hypothesis, gcd(a; ni "" 2 - 1 , a:"2 - 1 ) = xn-1 in F p m [a;],(x n -l)\(x n i "" 2 - 1 ) in R[x], and (a;n - l)|(a; n2 - 1) in R[x]. But gcd(a;ni - l,a; n2 - 1) = gcd(a;ni - 1 - ar" 1 "" 2 ^" 2 - l),a; n2 - 1) = g c d ^ " 1 " " 2 - 1, x" 2 - 1) in Fpm [a;]
253
Therefore gcd(x ni - l,xn2 a ; n i - n 2 ( x n 2 _ 1)
+
- 1) = xn - 1 in ¥pm[x].
(a;«i-»2 - 1 ) , ( x " - l ) | ( : c
ni
- 1) in
Since xni - 1 = R[x].
Proposition 3. Let g(x) be a monic polynomial over Fp™ without multiple roots and x J( g(x) in Fp™[a;]. Then g(x) has a unique Hensel lift in R[x]. Proof. By Proposition 1, g(x) has a Hensel lift in R[x\. Let / (1) (a;) and f{2){x) be two Hensel lift of g(s) in R[x\. Then both of them are monic polynomials over R, f^(x) = f^(x) = g(x), and there are positive integers ni and n2, both not divisible by p, such that /^1^(a:)|(a:"1 — 1) and /( 2 )(a;)|(x n2 - 1), respectively, in R[x]. Thus g{x)\(xni - 1) and g(x)\(xn2 — 1) in Fp™[a;]. We distinguish the following two cases. (a) nx = n2. For simplicity, let n = ni = n2. Assume that (5) is the unique factorization of xn — 1 into a product of distinct monic irreducible polynomial gi(x)(i — 1,2,-•• , r) over Fpm, that (6) is a factorization of xn — 1 into a product of pairwise coprime monic basic irreducible polynomial fi(x)(i = 1,2, •• • ,r) over R such that (7) holds, and that (8) is the unique factorization of g(x) into a product of distinct monic irreducible polynomial gi(x)(i — 1,2,•• • , s, where 0 < s < r) over Fpm. By the proof of Proposition 1, the monic polynomial f(x) defined by (9) is a Hensel lift of g(x) . Since f^(x) — g(x) and (8) is a factorization of g(x) into a product of distinct monic irreducible polynomials over Fp™, by Hensel's Lemma there are monic polynomials hi(x),h,2(x), • • •, hs(x) e R[x] such that fW(x)=h1(x),h2(x),---
,hs(x),
~hi(x) = gi(x),i = 1,2,-•• ,s, and h\{x), h2(x), • • • ,hs(x) are pairwise coprime in R[x]. Then hi(x),h2(x), •• • ,hs(x) are pairwise coprime monic basic irreducible polynomials over R. Since fM(x)\(xn - 1) in R[x], by the unique factorization of xn - 1 over R (see [3] or [6]), all hi(x),h2(x), ••• ,hs(x) appear in {/i(a;),/2(a;),--- ,fr(x)}. Since hi(x) = g^x) = fi(x),i = 1,2, ••• ,s, and g\{x), g2(x), • • •, j s (x) are distinct, we must have hi(x) = fi(x),i = 1,2, • • • ,s. Consequently, f^(x) = f(x). Similarly, f^(x) = f(x). (b) ni 7^ n2. Let n = gcd(ni,ri2), then p f\ n and, by Lemma 2, xn - 1 = gcd(x ni - l,xn2 - 1) in Fpm[a;], (xn - l)\(xni - 1) in R[x], and (xn - l)\(xn2 - 1) in R[x). We deduce g(x)\(xn - 1) in Wp™[x]. By the proof of Proposition 1, there is a monic polynomial f(x) € R[x] such that
254
f(x) = g(x) and f{x)\{xn - 1) in R[x]. Prom f(x)\{xn - 1) in R[x] and (xn - l)\(xni - 1) in R[x] we deduce f(x)\(xni - 1) in R[x]. By Case (a), f(x) = /(^(a;). Similarly, /(a;) = fW(x). This proves the uniqueness of the Hensel lift. 3. A n Algorithm to Compute the Hensel Lift Based on Propositions 1 and 3 of the proceeding section we formulate the following algorithm for computing the Hensel lift of a monic polynomial over Fpm in R[x]. Algorithm A Given a monic polynomial g(x) of degree > 0 over FP"» to compute the Hensel lift of g(x) in R[x] we proceed in the following steps. (1)
Test whether x\g{x) in Fp™[a;]. If yes, we are finished and g(x) has no Hensel lift in R[x]. If no, go to step 2. (2) Compute gcd(g(x),g'(x)) and let it be d(x). If degd(a;) > 0, we are finished and g(x) has no Hensel lift in R[x\. If Aegd{x) — 0, go to step 3. (3) Factorize g(x) into a product of distinct monic irreducible polynomials over Fp™ by Berlekamp's Algorithm. Let the result be g{x)
=g1(x)g2{x)---gr(x),
where gi(x),g2(x), •• • ,gr{x) are distinct monic irreducible polynomial over Fpm. Let deg<7i(a;) = rii,i = 1,2, ••• , r and go to step 4. (4) Compute lcm[p mni - l,pmn* - 1, • • • , p m n - - 1]. Let the result be n, then p does not divide n and g(x)\(xn — 1). Go to step 5. (5) Divide xn — 1 by g(x) by division algorithm. Let the quotient be gi(x). Then xn — 1 = g(x)gi(x) and gcd(g(x),gi(x)) — 1. Go to step 6. (6) By the constructive proof of Hensel's Lemma construct two coprime monic polynomials f(x), fi(x) 6 R[x] such that xn — 1 = f(x)fi(x) in R[x] and f(x) = g(x), f1(x) = gi(x). Then f(x) is the Hensel lift of g(x) in R[x]. Now we state Step 6 of Algorithm A in details as Algorithm B. Algorithm B Given a monic polynomial / G R[x] and a factorization of / , / = gxg2 in Fpm[x], where g\ and g2 are coprime monic polynomials
255 € Fpm[:r], to compute two coprime monic polynomials / i and 7*2 £ R[x] such that / = /1/2, /1 = pi and j*2 = 92 we proceed inductively as follows. (1)
Let /{ ' and / 2 be monic polynomials € R[x] such that /> ' = = 1,2. Then / - f™f™ e (p). Compute (/ - f[1]f^)/p. Let the result be fcW. Then fcW € i?[a;],degA;W < deg/, and
9i,i
/ = /i ( 1 ) / 2 ( 1 ) + Pk{1) in R[x}. (2)
Let n > 1 and assume that monic polynomials /^ and / 2 £ i?[x] and a polynomial k^ £ R[x] of degree < d e g / have been constructed such that / = /(")/(") + p 2 - 1 f c ( n ) i n i ? [ a . ] and
7 F = 9i,* = l,2. (a)
By Algorithm C below we find polynomials \[n', A2 G i?[a;] of degrees < deg/ 2 and deg/^ , respectively, such that A ^ A(n) + A2n) / 2 (n) = *(">
(b)
Compute f[n) +p 2 " _ 1 A^ n ) , &n) + p 2 " - 1 A
/ P y = 3i,i = l!2. Step 2 will stop when n = I + 1 where / is the positive integer such that 2' > e > 2 ' - 1 . (3) L e t / i = / f , + 1 ) , t = l,2. Algorithm C Given two coprime monic polynomial /1 and J2 over R and a polynomial fc over R with deg fc < deg f\ + deg f% to compute two polynomials Ai and A2 over R such that A1/1 + A 2 / 2 = k, and deg Ai < deg / 2 , deg A2 < deg / 1 , we proceed as follows.
256
(1)
By Euclidean algorithm find a i , a 2 E Fp™[x] such that a i / x + a 2 / 2 = 1. Let ai be an original of of and a2 be an original of o j under the homomorphism (4). Go to Step 2. (2) Divide a\f\ + a2f2 - 1 by p. Let the result be g € R[x]. Go to Step 3. (3) Compute YnZii-PQY- L e t t h e r e s u l t b e L T h e n (l + l)ka1f1 + (l+l)ka2f2
= k.
Go to Step 4. (4) Divide (l + l)kai by / 2 by division algorithm and divide (l + l)ka2 by / 1 . Let the result be (1 4- l)kai = qif2 + Ai, where qi,\i
€ R[x] and degAi < d e g / 2 ,
and (1 + l)ka2 = q2fi + A2, where g2, A2 € R[x] and deg A2 < d e g / i . T h e n A 1 / 1 + A 2 / 2 = A;. Finally we remark that when Fpm = F 2 and R — Z4, the Hensel lift of a polynomial g (x) over F 2 without multiple roots and not divisible by x can be calculated by using Graeffe's method for finding a polynomial whose roots are the squares of the roots of g(x), see [4] and [6]. References 1. Bonnecaze, A., Sole, P, and Calderbank, A. R., Quaternary quadratic residue codes and unimodular lattices, IEEE Trans. Inform. Theory 41(1995), 366377. 2. Krull, W. Algebraische Theorie der Ringe, Math. Ann. 92(1924), 183-213. 3. MacDonald, B. R. Finite Rings with Identity, Marcel Dekker, 1974. 4. Uspensky, J. V. Theory of Equations, McGraw-Hill, 1948. 5. Wan, Z.-X. Introduction to Abstract and Linear Algebra, Studentlitteratur, Lund, Sweden/ Chatwell Bratt, Bromley, UK, 1992. 6. Wan, Z.-X. Quaternary Codes, World Scientific, Singapore, 1997.
Some Advances Related Particle Systems —Estimation for Eigenvalue and N e w ^>24 Field
YAN SHI-JIAN
Department of Mathematics, Beijing Normal University 100875, Beijing, P.R. of China
This note introduces some advances which are related interacting particle systems and obtained by our group. Mainly the estimation for eigenvalue by coupling method was obtained by Chen Mu-Fa and Wang Feng-Yu and a new convergent lattice approximation for the (/>24-quantum field was constructed by Zhou Xian-Yin. I Application of coupling method to estimate the first eigenvalue Chen Mu-Fa have developed the coupling method of stochastic processes since he studied interacting particle systems. The optimal coupling w.s.t. to distance p was first proposed by Chen and then applied to estimate the first (nontrivial) eigenvalue of Laplacian on Riemannian manifolds by Chen and Wang Feng-Yu. This is a famous classical research problem of many branches in Mathematics, including the ergodic theory of stochastic processes and there are a lot of literatures. Denote by d, D and K respectively the dimension, the diameter and the lower bound of Ricci curvature (RicciM > Kg) of the manifold M. The problem is to estimate Ai in terms of these three geometric quantities. For an upper bound, it is relatively easy since the classical variational formula
X1 = mf{fM IIV/II2 : / € C\n(f)
= 0,TT(/ 2 ) = 1}
can be applied. However, the lower bound is much harder. The previous works have studied the lower estimates case by case by using different elegant methods, eight of the most beautiful lower bounds are listed in the following table. Supported by NFSC
257
258
A .Lichnerowicz (1958)
d-lK>
P.H.Berard, G.Besson & S.Gallot (1985) P.Li &S.T.Yau (1980) J.Q.Zhong&H.C.Yang (1984)
K
(1)
^
1>0, (2)
\fQD/2cos^tdt) IT2
(3) K
^
>^
(4)
-l-^l-4D2K(d-l)
P.Li&S.T.Yau (1980) K.R.Cai (1991) H.C.Yang(1989)& F.Jia(1991)
D*(d-1) ^ 7T
+ K,
K
^n
(5)
-
K<0
(6)
2
7^e-a, 2
H.C.Yang(1989)&F.Jia(1991)
'
^e-
a
',
d>5,
(7)
K<0.
2
K<0
(8)
where a = Dy/\K\(d-l)/2, a' = Dy/\K\((d-l)V2)/2. All together, there are five sharp estimates ((1),(2),(4),(6) and (7)). The above authors including several famous geometers and the estimates were awarded several times. ^From the table, it follows that the picture is now very complete, due to the effort by the geometers in the past 40 years. The original starting points of Chen is to learn from geometers and hope to apply their methods and results to the phase transitions of interacting particle systems. It is surprising that they actually went to the opposite direction, that is, studying the first eigenvalue by a probabilistic approach and finally unexpected that they find a general formula for lower bound. Their main result is Theorem [General formula] (Chen & Wang (1997))2. Ai > sup inf ferre(o,D)
4/(r) fcC{s)-lds$°C{u)f{u)du
where T = {/ e C[0, D] : / > 0 on (0, £>)} and C(r) = c o s h ^ 1 re
~K 2Vd-l
(0,D).
It is a dual of the classical variation formula and there are no common points in these two formulas. This explains the reason why such a formula never appeared before. Certainly, the new formula can produce a lot of
259
new lower bounds. For instance, let a be the same as above and let /? = 7r/2D, applying the formula to the test functions sin(/3r), sin(ar), sin(/3r), coshd~ 1 ( a r ) sin (/3r) successively, we obtain the following: Corollory (Chen &Wang (1997))2.
A l
> g
+ m a
x{^,l-^}x,
K>0;
^ ^ { l - c o s ^ ^ ] } " ,
(9) d>l,*>0;
(10)
2
A i > ^ + ( | - l ) ^ ,
K<0;
(11)
Remark 1. The corollary improves all the estimates list above (l)-(8). (9) improves (4);(10) improves (l)and (2); (11) improves (6); (12) improves (7) and (8). 2. The theorem and corollary valid also for the manifolds with convex boundary with Neumann boundary condition. In this case, the estimates (l)-(8) are believed by geometers to be true. However, only estimate (1) was proved until 1990. Except this, the others in (2-(8)(and furthermore (9)-(12)) are all new in geometry 2 . 3. For more general non-compact manifolds, elliptic operators or Markov chains, Chen and Wang also have the corresponding dual variational formula' 4. The above study was the first time for applying the probabilistic method to estimating the eigenvalues. For a along time, almost nobody believes that the probabilistic method can achieve sharp estimate. From these facts, the influence of the above results to probability theory and spectral theory should be clear. Chen and Wang also have studied the closely related logarithmic Sobolev inequality, Nash inequality, Poincare equality and so on 4 . Chen also have studied Liggett inequality 5 . Wang have studied new type of Harnark inequality and general type of Poincare-Sobolev inequality 6,7 ' 8 . II. (f>24 quantum fields and polymer measures The <^d4 —fields is formally described by the following probability measure:
260
H N
^ ^
d(t>(x)expi-
f[\i\V4>(x)\2
+ \2
X€R
(13) where Nxlt\2t\3 is the normalization constant, A& > 0, k — 1,2,3 are real coupling constants and (j>{x) is a real valued field. There have been many approaches to the problem of giving a meaning to the above formal measure for d = 2,3. One way(heuristically) is that the integrations in (13) can be approximated respectively by ad~2 T,\x-y\=a:x,yeaZd(^-(f>y)2, ad T,xeaZd <^2 d d 4 as a d d and a J2X£aZ ^ "* 0,where aZ = {ax : x € Z }, and thus (13) can be approximated by corresponding lattice <j>44 fields(e.g.the case a1 = a of (14) in the below). Along this approach,one can rigorously construct a continuum fid4 for d = 2,3. In a paper of S.Albeverio and Zhou Xian-Yin(1995), a new lattice approximation to (13) was considered. They choose a "new cut-off" a' = a'(a) > a satisfying limo_+0+ a1 (a) — 0,a'Zd C aZd. Heuristically, the quantity f
x£aZd
\x—y\=a:x,%
-(X2ad+4dXiad-2) J2 ^ 2 -A 3 (a') 4 £ t£aZd
xea'Zd
4>A, '
(14)
where iV0i0/is the normalization constant. Denote the measure (14) by vaAi, This model can be thought of as a lattice 4^ield with different lattice cutoffs in the free and interacting parts. Similarly as for va,a, they gave a rigorous sense to i/ a , a '. They proved in that paper that 1 0 {va,a} and {va,a'} have different asymptotic behaviors, if d = 2 and a1 satisfied the conditions: limsup |loga| 5 ' 4 a' < +00, a->0+
liminf | loga| 2 a' = +00. a
~>0+
Before this work, they had studied the corresponding problem for the discrete Edwards model in two dimensions9. The idea of this work came from there. Polymer.Let {Btd} be the standard Brownian motion in Rd, the socalled polymer measure {Edwards model} on Co ([0,1] -> Rd) is formally
261
denned by vg{
f
j
S(Bsd - Btd)dsdt\
fi(dw),
where g > Ois a coupling constant, and Ng is the normalization. For d < 3, the existence of iy9has been proved by a number of authors, and the perfect results was recently obtained by S.Albeverio, E.Bolthausen and Zhou Xian-Yin 11 . It is difficult to give a strict definition for d = 4. It same as the 0 4 -quantum fields, d — 4 is critical and very interesting case. In fact, M.Aizenman, J.Frohlich had respectively proven that d4^ield, d > 5 is Gaussian. By applying a new method, S.Albeverio and Zhou X.Y. obtained the sharp estimations of random walk and Brownian motion in four dimension for all cases 12 ' 13 . Before that work, all approaches which study the intersection of random walk or Brownian motion in four dimensions cannot solved the problem, except G.F.Lawler obtained some sharp estimations for special cases. As an application of this results, they discussed a modified Domb-Joyce model in four dimensions, it formally approximate (continuum) Edwards model 14 . They obtained the tightness for some cases; the probability measure from the tightness can be expected as polymer measure in four dimensions. These woks may be a important significant explore for the existence of polymer measure in four dimensions. Unfortunately, Dr. Zhou cannot continue his ambitious significant research process since he died.
IH Sketch of t h e M a i n P r o o f of I Here we adopt the analytical language. Given a self-adjoint elliptic operator L, denote by {Tt}t>o the semigroup determined by L: Tt = etL. The coupling {Tt}t>o of {Tt}t>o and {Tt}t>0 simply means the following marginality: ftf(x,y)
= Ttf(x) (resp. ftf(x,y)=Ttf(y))
(1)
d
for all / € C%(R ) and all (x,y) (x ^ y), where on the left-hand side, / is regarded as a bivariate function. Corresponding to {Tt}t>o, we have L. Step 1. Let g be the eigenfunction of — L corresponding to Ai. We have jTt9{x)
= TtLg(x) =
-XMx).
Hence for fixed g and x, we obtain Ttg(x)=g(x)e-^.
(2)
262
Step 2. Consider compact space, g is Lipschitz with respect to Riemannian distance p. cg: Lipschitz constant. Now, the m a i n condition we need is the following: ftp(x,y)
xfy
= g(x), g2(x,y) = g{y), e-^\g(x)
- g(y)\
= \Ttg(x)-Ttg(y)\
(by (2))
= \Ttgi(x,y)-ftg2(x,y)\
(by(l))
(Lipschitz property) (by (3)).
Hence Ai > a. REFERENCES 1. M. F. Chen, Estimation of spectral gap for Markov chains, Acta Math. Sin.(New Series) 12 (1996), 337-360. 2. M. F . Chen, F.Y.Wang, General formula for lower bound of the first eigenvalue, Sci.Sin.(English Edition) 40 (1997), 384-394. 3. M. F . Chen, F.Y.Wang, Estimation of spectral gap for elliptic operators, Trans. Amer. Math. Soc. 349 (1997), 1239-1267. 4. M. F. Chen, Eigenvalues, inequalities and ergodic theory, Chinese Sci Bulletin(English Edition) 45:9 (1999), 769-774. 5. M. F. Chen, Y. Z. Wang, Albraic convergence of Markov chains (2000), to appear Ann. Appl. Prob... 6. F. Y. Wang, Harnack inequalities for log-Sobolev functions and estimates of logSobolev constant, Ann. Probab. 27 (1999), 653-663. 7. F. Y. Wang, Sobolev type inequalities for general symmetric forms, Proc. of AMS 128:12 (2000), 3675-3682. 8. F. Y. Wang, Functipnal inequalities, semigroup properties and spectrum estimates, Inf. Dim. Anal., Quan. Prob. Rel. Topics 3:2 (2000), 263-295. 9. S. Albeverio, X. Y. Zhou, A new discrete Edwards model and a new polymer measure in two dimensions, Comm. Math. Phys. 177 (1996), 469-506. 10. S. Albeverio, X. Y. Zhou, A new convergent lattice approximation for the 4>\ — quantum fields, SFB 237-Preprint Nr.284 Institut fur Mathematik Ruhr-UniversitatBochum October 1995. 11. S. Albeverio, E. Bolthausen, X. Y. Zhou, A modified discrete Edwards model in three dimensions, Preprint 1995. 12. S. Albeverio, X. Y. Zhou, Intersection of random walks and Weiner sausages in four dimensions, Acta Appl.Math. 45 (1996), 195-237.
263 13. S. Albeverio, X. Y. Zhou, Intersection properties of Brownian motions in four dimensions, Ann.Inst.Henir Poincare. 14. S. Albeverio, X. Y. Zhou, A modified Domb-Joyce model in four dimensions, Markov Processes Rel. fields 2 (1996), 369-400.
A N O T E ON LOCALLY REAL H Y P E R B O L I C SPACE W I T H FINITE VOLUME
YI-HU YANG* Department of Applied Mathematics, Tongji University, Shanghai, China
In this note, we will prove that any finite volume quotient of real hyperbolic space of dimension at least 3 is not properly homotopical equivalent to a quasi-projective variety.
1. Introduction The purpose of this short note is to show the following Theorem 1 Let M be a compact Kahler manifold, D be a normal crossing divisor of M. Denote M\D by M. Assume that T is a nonuniform lattice of SO(n,l)(n > 3), i.e., T \ SO(n,l)/SO(n) is noncompact and of finite volume with respect to the standard symmetric Riemannian metric. Then M is not properly homotopical equivalent to T \ SO(n, l)/SO(n). R e m a r k s : a) In the compact case, J. Carlson and D. Toledo 3 proved that any cocompact lattice of SO(n, l)(n > 3) cannot be the fundamental group of a compact Kahler manifold; b) if M is a quasi-projective variety, by Hironaka's theorem 5 , topologically, M is just a smooth projective variety deleting a normal crossing divisor; c) the condition of n > 3 is obviously necessary. The idea of the proof is to use harmonic map theory. We assume that the theorem is not true, namely, M is properly homotopical equivalent to N = T \ SO(n,l)/SO(n). First, we choose a complete Kahler metric of finite volume on M. Then, we construct a harmonic homotopic equivalence of finite energy from M (under the metric constructed) to N using the theory due to Jost-Yau and Jost-Zuo 6 ' 7 , 8 . The generalization of the famous argument of Siu and Sampson 12 ' 10 to the noncompact case then gives a "the author supported partially by nsf of china (no. 19801026)
264
265
strong restriction to this harmonic map. But, this will be a contradiction to our assumption at the beginning of the proof. The first version of this paper was completed when the author was visiting ICTP, Trieste during 19992000 and appeared as a preprint of ICTP titled by " A note on nonuniform lattices of SO(n, 1)". He thanks the institute for hospitality. 2. The construction of harmonic maps of finite energy Let M be a compact Kahler manifold with a fixed Kahler metric UJ, D be a fixed divisor with (at worst) normal crossing condition and D = Dsi=1Di be a disjoint union of connected components of D. Di consists of some irreducible components D],D2,--- , D™' (i = 1,2, ••• ,s). Let N = T \ SO(n, l)/SO(n) be a noncompact quotient of finite volume of real hyperbolic space of dimension at least 3. Assume M = M \ D is properly homotopical equivalent to N. We will construct a harmonic homotopic equivalence from M to N under an approriate metric (which we will construct in the following) on M and the standard metric of N. Let a\(i = 1,2,-- ,p;j = 1,2,--- ,m,i) be a defining section of D\ in 0(M, [Dj]), which satisfies \aj\ < 1 under certain Hermitian metric of [Dj] and defines a coordinate system in a small disk transversal to D\. Set Cj = a\
1S a
w h e r e <j> suitable C°° cut-off function on [0, oo) so that <j){s) is identical to one on [0, e) and to zero on [2e, oo) for sufficiently small e > 0 and c is taken sufficiently large so that g is positive definite. Then g is a Kahler metric. Furthermore, one can show that (M, g) is complete and of finite volume 2 ' 7 , 8 . Since .ZV is a finite volume noncompact quotient of real hyperbolic space under the nonuniform lattice T, it is well known that iV can be simply compactified by adding some isolated points to its cusps. In particular, near each cusp, it is a topological product of R+ and a torus and its metric is of form: dp + e~2pdu)2, where du2 is the flat metric of the torus. Before going on our construction, we recall a notion due to Jost and Zuo 8 (for the noncompact curves case, also see n ) . A loop 700 in M around D is called small if there exists a homotopy H : [0,1] x [0,1] -> M with
266
H(.,0) = 7oo, H([0,1] x [0,1)) C M and H(., 1) is a point on D. Thus, 700 is freely homotopic in M to arbitrarily short loop under the induced metric or the above constructed complete metric. Definition. Let G/K be a symmetric space of noncompact type. A homomorphism p : 7Ti (M) —> G is called stabilizing at infinity if for every small loop 700 around a connected component of D, inf
dist(z, pij^z)
= 0.
z€zG/K
Remark, pijoo), if nontrivial, is strictly parabolic or elliptic. In our present case, it will be strictly parabolic, as we will see. In 8 , Jost and Zuo obtained Theorem 2 Let M be as above with the above constructed metric. Assume that the representation p : 7rj (M) —> G is stabilizing at infinity and reductive. Then there exists a p-equivariant harmonic map u : M —>• G/K from the universal covering of M to G/K with finite energy on a fundamnetal domain of M. Now, we turn back to our construction. Since M is properly homotopical equivalent to iV, one can get a special homomorphism p : 7Ti (M) = T C SO(n, 1), which certainly satisfies the conditions of Jost-Zuo's theorem except stabilizing property at infinity, which we will show in the following. Take a splitting of M = Mb{JU?=1Mi, here Mb is bounded in M, M, is a neighborhhod of -Dj. Obviously, any small loop 700 around D can be assumed to lie in one of Mi,i = 1,2,••• ,p. Since M is properly homotopic to N, corresponding to this splitting, we also have a splitting of N = Nf,\J U?=1Ni, here iVj is bounded in N, Ni is a neigborhood of a cusp of N. The homotopy maps Mb into Nf,, Mt into Ni,i = 1,2, • • • ,p. So, if a small loop 700 lies in Mi and represents a nontrivial element of -KI(M), p(loo) a l s o represents a nontrivial element of T, and can be represented by a loop lying Ni. Since p(7oo) is nontrivial and lies in Ni, by a point compactification of Ni, one has that /o(7oo) is strictly parabolic. So, one obtains that p is stabilizing at infinity. By Jost-Zuo's theorem, we get a harmonic map of finite energy from M (under the constructed metric) to N, which induces p, and hence is a homotopic equivalence. Remark. In 8 , Jost and Zuo first constructed a finite energy map, then they used a variational technique to get the above harmonic map. In fact, if applying the construction of this finite energy map to our present setting, it is easy to see that this map is actually proper and one can also get the above harmonic map using Schoen-Yau's technique 13 . Then the same reasoning as in section 3c of 6 derives that this harmonic map is essentially proper.
267 Summing up the above arguments, we have the following Lemma 1 Let M and N be as above. Then there exists a properly harmonic homotopic equivalence of finite energy from M to N. 3. Some properties of the harmonic maps The argument in this section is the generalization of that of Siu and Sampson in the compact case to the noncompact case. Our exposition is general, which obviously includes our present case. Let M be a complete Kahler manifold with a fixed Kahler metric u>, N = T \ G/K a locally symmetric space of noncompact type. Here G is a semisimple Lie group, K is a maximal compact subgroup of G, and Y is a lattice of G (for standard references see 4 ' 9 ) . Let g be the Lie algebra of G and t the Lie algebra of K, Then one has the Cartan decomposition 0 = t + p such that [p,p] C t, [t, p] C p. Denote by pc the complexification of p. One can then identify the complexification of the tangent space at any point of N with pc. This is unique up to the right action of K and the left action of V. Since these actions preserve all the relevent structures, we may regard df (T^'°M) as a subspace of p c , for any map / : M -> N and any point x £ M. On the other hand, in general, let h : M —> TV be a smooth map from a Kahler manifold to a Riemannian manifold. If we denote K,w2,--_,wm,u;r, (J2,--- ,u>m} to be a local orthonormal coframe of M and {fl 1 ,^ 2 ,--- ,8n} to be a local orthonormal coframe of TV, then one can define hxa and h^ by the equation
for all 1 < % < n, 1 < a < m, and 1 < a < m. As usual, one can also define the higher order covariant derivatives hla~, hl -j, / i ^ , /li-^, hlagy, /i^g-y) hl -p., etc. And one has the commutation formulae concerning higher order covariant derivatives, for example,
a0y ~
<*7/3
=
~-™-iklm'lah-ph,^
,
in the second formula, we used Kahler condition. Using the above notations, the harmonic map equation is then given by
268
We say that h is pluriharmonic if it satisfies ap
for all i, a, /?. ^From now on, we assume that h : M —> N is a harmonic map with finite energy, here M is a complete Kahler manifold with a fixed Kahler metric w and N is a locally symmetric space of noncompact type. Fix a point O € M. For any positive integral number, take a C°° cut-off function rjk : M ->• R+ as follows: rjk =
l,onB0(Rk), =0,onM/B Vk 0{2Rk), 2 |Vr?fc|w < — ,onM, rik
here, {#&} is a sequence of positive numbers, which is strictly monotone and divergent. One then has
Integrating both sides on M and using Holder inequality, one has
/ vlKpi2 = -2 / w i f c ^ X - / 4^00 JM
JM
JM
<2e[ Vl\h^ + - [ k,/KI 2 ~ [ vlh^hl c
JM
JM
JM
As e is sufficiently small, say e < | , one has 0 < (1 - 2e)
t
/,W
JM IM
JM JM
PP
On t h e other hand, using t h e commutation formulae a n d t h e hermitian negative curvature condition of TV under Sampson's sense 10 , one h a s a/3/3 <*
W "a = C1^ = >0.
Riklmhphahj)hla
-Riklmhlahphwh^
269 Thus, one has
0<(l-2e)/
4K
JM
<
2 •/ e JMM C C
( )
R\
\v^\2K\2 IM|2 [/JBo(2R ) K\
JBo(2Rk)k
where c(e) is a positive constant depending only on e. As k —> oo, by the choice of rjk and the finiteness of energy of h, one has hl -, = 0, i.e., h is pluriharmonic. By the above argument, one also has, for any k,
JM =
~
Vk^-iklmh^hgh-^h^. P
JM
Since the integrand of the right-hand side integral is nonnegative, so, one has on M
Riklmhih^hf
= 0.
On the other hand, RikimKhk0hLh%
= < R{dh{ea),dh{ep))dh{ea),dh{^) = - < [dh(ea),dh(ep)],\dh(ea),dh(ep)]
here
ei,e2, • • • ,em,ej,
(jjl,u)2,-
• • ,wm,w1,o;2,
e^, • • • ,em
are
the
dual
> >, frames
of
• • • ,u>m. So, [dh(ea),dh(ep)} = 0 for all a, /3. Thus, one has that if one identifies dh{T^'0M) with a subspace of p c , then dh(TlfiM) is an abelian subspace of p c . Therefore, dimcdh(T^'0M) should not be greater than the rank of N. If one applies this assertion to our case, one obtains Lemma 2 Let h : M -> N be the harmonic homotopic equivalence of finite energy in the last section. Then h has rank at most 2. 4. T h e proof of t h e t h e o r e m Assume that M is properly homotopical equivalent to N = T \ SO(n, l)/SO(n), n > 3. Therefore, by the section 2, one has a harmonic homotopic equivalence of finite energy h from M (with the constructed metric) to N (with the standard symmetric metric); by the section 3, this
270
harmonic homotopic equivalence has rank at most 2. Now, we will proceed to derive a contradiction. Assume that TV has at least two cusps. It is easy to see that the boundary of a suitable neighborhood of each cusp represents a nontrivial homology element in Hn~x (TV, R); by Lemma 1, one also sees that the above harmonic map is proper. So, there exists at least some point at which this harmonic map is of maximal rank n. This is a contradiction (Note that if one assumes n > 4, we need not consider the properness of the harmonic map). Therefore, the proof is reduced to the case that TV has only one cusp. We will use a result of Selberg * to reduce this case to the case of at least two cusps. More precisely, assuming that TV has only one cusp, we will make a finite covering TV' of TV, which is of at least two cusps, using Selberg's result. Then, we make the corresponding covering M' of M and lift the above harmonic map to another harmonic map h' : M' —> TV' with respect to the corresponding lifting metrics. It is not difficult to see that h' satisfies all properties of h namely, h' is a properly harmonic homotopic equivalence of finite energy from M' to TV'. So, the above argument also implies that h! is of maximal rank at some point of M'. Since h' is the lifting of h, So, h also has maximal rank at some point of M. The contradiction is obtained. Assume that TV has only one cusp. Since TV is of finite volume, T is not a parabolic subgroup of SO(n, 1), namely, there exists an element 7 of T, which does not fix any lifting of the unique cusp of TV to the boundary at infinity of SO(n, l)/SO(n). On the other hand, a result of Selberg * says that any lattice of SO(n, 1) is residually finite, that is to say, the intersection of all subgroups of finite index of a lattice is its unit. In particular, so is T. Therefore, we can choose a subgroup of finite index of T, which does not contain the above element 7. A lifting argument shows that one can actually choose a normal subgroup of finite index. Corresponding to this normal subgroup, we have a smooth finite covering of TV, which has at least two cusps thanks to the properties of 7. This completes the proof of the theorem. References 1. H. Bass, Introduction to Some Methods of Algebraic K-theory. CBMS-NSF reg. Conf. Ser. Appl. Math., 20(1974). 2. M. Cornalba and P. Griffiths, Analytic cycles and vector bundles on noncompact algebraic varieties. Invent. Math., 28(1975), 1-106. 3. J. Carlson and D. Toledo, Harmonic mappings of Kahler manifolds to locally symmetric spaces. Publ. Math. IHES, 69(1989), 173-201. 4. S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces. Academic Press, 1978.
271 5. H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero, I, II. Ann. Math., 79(1964), 109-326. 6. J. Jost and S.-T. Yau, The strong rigidity of locally symmetric complex manifolds of rank one and finite volume. Math. Ann., 275(1986), 291-304. 7. J. Jost and S.-T. Yau, On the rigidity of certain discrete groups and algebraic varieties. Math. Ann., 278(1987), 481-496. 8. J. Jost and K. Zuo, Harmonic maps and Sl(r, C)-representations of fundamental groups of quasi-projective manifolds. J. Algebraic Geometry 5(1996), 77-106. 9. M. S. Raghunathan, Discrete Subgroups of Lie Groups. Ergebnisse der Mathematik und ihrer Grenzgebiete 68, Springer (1972). 10. J. Sampson, Applications of harmonic maps to Kahler geometry. Comtemp. Math., 49(1986), 125-134. 11. C. Simpson, Harmonic bundles on noncompact curves. J. Amer. Math. Soc, 3(1990), 713-769. 12. Y. T. Siu, The complex analyticity of harmonic maps and the strong rigidity of compact Kahler manifolds. Ann. Math., 112(1980), 73-111. 13. R. Schoen and S.-T. Yau, Harmonic maps and the topology of stable hypersurfaces and manifolds of non-negativeRicci curvature. Comment. Math. Helv., 39(1976), 333-341.
INTERESTING PROPERTIES OF THE SETS: AT 2 [l 2 ,2 2 ,3 2 ,...],iV3[l 3 , 2 3 ,3 3 ,...] AND JV 4 [1 4 , 2 4 , 3 4 ,--] Yanqian YE Department of Mathematics, Nanjing University, Nanjing 210093, P. R. China Abstract This paper deals with many properties of the sets N2,N3,N± and relations between them, chiefly, the possibility of expressing an element in Ni as the sum of some different elements in Nj (i may equal to j).
Our interest arose first from R. Norrie's identity 304 + 1204 4- 2724 + 315 4 = 353 4
(1)
and asked: whether 353 in (1) can be made smaller? We noticed that integers in JV4 under the decimal system have last digits 1,6,1,6,5,6,1,6,1,0
(2)
consecutively and recurrently. Moreover, if we set x4 = abc- • -hijk, then (i) When k is odd, j must be even; (ii) When k = 5, we have hijk = 0625; (iii) When k is even, j must be odd; (iv) When k = 0, then hijk = 0000. By using these properties we proved in [1] that the equation x4 + y4 + z4+w4
= u4
(3)
has no solution in integers with u < 200. Then in [2] we gave the following definitions: Definition 1. If n 4 cannot be expressed as the sum and difference of some smaller but all different numbers in JV4, it is called an additive prime number. E x a m p l e s . 1 4 ,2 4 , • • • ,8 4 ,10 4 are prime numbers. 272
273
Definition 2. If n 4 can be expressed as the sum of some smaller but all different numbers in N4, it is called an addtitive composite number. Examples. 154 = 144 + 94 + 8 4 + 64 + 4 4 , 25 4 = 21 4 + 204 + 124 + 104 + 8 4 + 6 4 + 2 4 are additive composite numbers. Definition 3. If n 4 is neither an additive prime number nor an additive compositive number, it is called an additive quasi-prime (or quasicomposite) number. Examples. 94 = 8 4 + 7 4 + 3 4 - 2 4 - l 4 , l l 4 = 104 + 8 4 + 5 4 + l 4 - 3 4 are quasi-prime numbers. Definition 4. If n 4 has both of the above expressions, it is still called an additive composite number. Example. 35 4 = 33 4 + 204 + 184 + 144 + 104 + 6 4 + 2 4 = 344 + 194 + 4 13 + 8 4 + 6 4 + 2 4 — l 4 is a composite number. For 1 < n < 100 we found in 7V4: (i) 9 prime numbers as shown before. (ii) 27 composite numbers: 15 4 , 25 4 , 35 4 ,•••, 95 4 ; 29 4 , 31 4 , 41 4 , 59 4 , 87 4 , • • •, and so on. (hi) 64 quasi-prime numbers: l l 4 , 12 4 , 13 4 , 144; 164, 17 4 , • • •, 24 4 ; 26 4 , 28 4 , ••• , and so on. Among which there may still be some composite numbers not discovered for the moment. Later on in [6], we have proved that For 1 < n < 200, there are only 23 quasi-prime numbers, they are 9 4 ,11 4 ,12 4 ,13 4 ,14 4 ; 16 4 ,17 4 ,--- ,24 4 ; 26 4 ,28 4 ,32 4 ,34 4 ,36 4 ,38 4 ,40 4 ,42 4 ,44 4 . So, the number of composite numbers is 200 — 9 — 23 = 168. We have stuided in [2] similar problem in the sets of squares N% — 2 2 2 [1 ,2 ,3 , • • •] and cubes: N3 = [1 3 ,2 3 ,3 3 , • • •], the results are comparatively easier to get, we have proved For l 2 < n 2 < 1002 in N2 there are 4 prime numbers: 1 2 ,2 2 ,3 2 ,4 2 ; 2 quasi-prime numbers: 6 2 ,8 2 ; all the rest 94 are composite numbers. For l 3 < n 3 < 1003 in N3 there are 6 prime numbers: 1 3 ,2 3 ,3 3 ,4 3 ,5 3 ,7 3 ; 3 quasi-prime numbers: 8 3 ,10 3 , l l 3 ; all the rest 91 are composite numbers. Conjecture. For n > 100 all numbers in N2,N$ and N± are composite numbers. If this Conjecture is true, then the introduction of the idea quasi-prime (or quasi-composite) has no signifigence.
274
In [3] we studied the possibility of even numbers in N2, N3 or JV4 to be expressed as the sum of some smaller and different odd numbers in the same set. We use 7V2° = [1 2 ,3 2 ,5 2 , • • •], JV3° = [1 3 ,3 3 ,---], and 7V4° = [ l 4 , 3 4 , 5 4 • • • ] to denote these latter sets. The results are different in different cases. (I) In N2 only 10 even numbers: 2 2 ,4 2 ,6 2 ,8 2 ,10 2 ,12 2 ,16 2 , 20 2 ,24 2 , 28 2 cannot be expressed as the sum of different elements in A^; all the rest even numbers can be expressed so. But the number of terms is 4 when 4 / n , it is 8 when 4|n. Examples. 142 = l l 2 + 72 + 5 2 + l 2 , 182 = 172 + 5 2 + 3 2 + l 2 , 322 = 2 19 + 172 + 132 + l l 2 + 72 + 5 2 + 3 2 + l 2 , etc., (II) In N3 only the first 12 even numbers cannot be expressed as the sum of different elements in N°, but in the expressions of the rest even numbers we cannot find a general role as in (I), the number of terms in the expressions may be 6,8,10 or 12. (III) In N4 there are 31 even numbers cannot be expressed as the sum of some different elements in JV|, they are 2 4 ,4 4 ,6 4 ,--- ,60 4 ; 64 4 . The expressions of the rest 19 even numbers all have 16 terms. Example. 624 = 47 4 + 45 4 + 35 4 + 33 4 + 31 4 + 294 + 274 + 254 + 23 4 + 4 21 + 154 + 134 + 74 + 5 4 + 3 4 + l 4 (In [3] we consider only the first 100 numbers). In [4] we studied the possibility of expressing a number in JV^ as the sum of some different elements in Nj,i ^ j , i,j = 2,3,4. (I) Relation between N3 and N4. They have infinitely many nmber of common elements: l 3 = l 4 , (2 3 ) 4 = 4 3 (2 ) , • • •, but no inclusion relation. We have proved that for 1 < n < 100, only 2 4 ,3 4 ,4 4 ,5 4 in N4 cannot be expressed as the sum of some different elements in JV3. Examples. 9 4 = 183 + 9 3 , 64 = 8 3 + 7 3 + 6 3 + 5 3 + 4 3 + 3 3 + 2 3 + l 3 (the expressions are not unique). Conversely, for 1 < n < 100, only 26 elements in N3 can be expressed as the sum of different elements in JV4, they are 43 3 ,49 3 ,55 3 ,59 3 ,61 3 ,62 3 , • • • , etc., Example. 62 3 = 174 + 164 + 13 4 + 124 + h i 4 (15 terms). (II) Relation between iV2 and N3. We have proved that only 2 3 ,3 3 ,4 3 in N3 cannot be expressed as the sum of some different elements in N2, but conversely, for 1 < n < 100, only
275
61 elements in N% can be expressed as the sum of different elements in N3, they are 3 2 ,6 2 ,10 2 , • • • , etc. As a special case we have got the well known identity: l 3 + 2 3 + 3 3 H + n 3 = [n(n + l)/2] 2 for all n. (Ill) Relation between 7V2 and N4 (N4 C N2). We have proved that for 2 < n < 100, every element in 7V4 can be expressed as the sum of different elements in N2, among which only 5 4 = 24 2 + 72 and 29 4 = 8402 + 41 2 contain the least (two) terms on the righthand side. Notice 5 and 29 are two important integers in the theory of Diophantine Equations. Because in 1966 W. Ljunggren has proved that a;4 — py2 — 1 for any prime number > 2 has solution in intergs only when p = 5 or 29. But conversely, for 1 < n < 100, only 58 2 ,63 2 ,94 2 ,98 2 can be expressed as 58 2 = 7 4 + 5 4 + 3 4 + 1 4 ,63 2 = 7 4 + 6 4 + 4 4 + 2 4 ,94 2 = 9 4 + 6 4 + 5 4 + 3 4 + 2 4 + l 4 ,98 2 = 9 4 + 7 4 + 5 4 + 2 4 + l 4 . For 100 < n < 200, only 8 elements in N2 : 106 2 ,138 2 ,153 2 ,166 2 ,178 2 ,190 2 ,191 2 ,193 2 can be similarly expressed. Moreover, we found there is a misprint in R.K.Guy's book: "Unsolved problems in number theory" Chap D, p.80. In which the sentence: "In fact even a 4 + &4 +
c
4
= d2
4
4
4
(4) 2
is unknown" is not true, since 12 + 15 + 20 = 481 was alredy known to Diophantus. We think that (4) should be ai + bi + ci = d3, where a, b, c are different. Finally, in [6] we refined the results in [2] and proved: (i) When n > 34 the expressions of (n + 1 ) 2 as the sum of some different smaller squares can be made such that the first term is n 2 . Examples. 35 2 = 342 + 72 + 4 2 + 2 2 , 27 2 = 26 2 + 7 2 + 2 2 , etc. Notice that 342 = 322 + 9 2 + 5 2 + 4 2 + 3 2 + l 2 cannot be refined to 34 2 = 33 2 + • • •. An equivalent statement of the above assertion is: For 34 < n < 100, 2n + 1 can be expressed as the sum of different squares. Moreover, it is easily seen that (ii) For n > 2, any n2 can be expressed as the difference of two squares. Examples 132 = 85 2 - 84 2 , 372 = 6852 - 684 2 , etc.. Similarly in N3 we have proved (iii) When n > 26, the expressions of (n+1) 3 as the sum of some different smaller cubes can be made such that the first term in n 3 .
276
Examples. 273 = 26 3 + 123 + 7 3 + 3 3 + 2 3 + l 3 , 193 = 18 3 + 103 + l 3 . Notice that 26 3 = 243 + 153 + 6 3 + 5 3 + 3 3 + 2 3 + l 3 cannot be refined to 26 3 = 25 3 + • • • . In -/V3 we have no statement corresponding to statement (ii) before. (iv) When n > 93 the expressions of (n + 1 ) 4 as the sum of some different quartics can be made such that the first term is nA. We have made a table given all partial sums of elements in N2 such that it will cover the whole latter part n > 129 in N, also the whole latter part n > 69 in TV0 = [1,3,5,•••] (see Statement (i)). Also Tables of partial sums < 2000 of elements n 3 in N3 and partial sums < 16000 of elements n 4 in N4 are obtained. We found these 3 tables have 12 common elements: 881, 882, 1296, 1297, 1313, 1377, 1378, 1393, 1394, 1568, 1649, 1650. Prom these numbers we can get some identities between elements of N2, N% and JV4. Examples. 881 = 4 4 + 5 4 = 9 3 + 5 3 + 3 3 = 29 2 + 62 + 2 2 , 1313 = 64 + 2 4 + l 4 = 9 3 + 8 3 + 4 3 + 2 3 = 362 + 4 2 + l 2 . Remark. Notice that our program differs from the theory of Waring's Problem in two respects: (a) Members in any expression are asked to be different from each other; (b) We do not confine the number of terms in all expreesions. Acknowledgments Project supported by the National Natural Science Foundation of China. References 1. Ye Yanqian, Some identities for the quartics of integers (to appear in J. Nanjing Univ., Math. Biquarterly, 2001). 2. Ye Weiyin and Ye Yanqian, A new kind of prime and composite numbers (to appear). 3. Ye Weiyin and Ye Yanqian, On the possibility for an even number in A?2, N3 or N4 to be expressed as the sum of different odd numbers (to appear). 4. Ye Weiyin and Ye Yanqian, Relations between numbers in Nz,N3 and Ni (to appear). 5. Yang Naikuo and Ye Yanqian, On the additive composite numbers in N4 (to appear).
6. Ye Yanqian, Interesting properties of additive composite numbers in N2,N3 and AT4 (to appear).
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List of Participants BAI Zhengguo (filEH) BOURGUIGNON Jean Pierre * CHANG Kung-ching (3K#&) * CHEN Hanfu ( & & & ) * CHEN Hanlin (|$$&ft) * CHEN Weihuan (WMU) * CHEN Zhihua (Mlt^) * CHENG Qingming (/ft^HJ!) * CHERN Shiing-Shen ( M i l ) * CHOW Yuan Shih (Mji$k) DING Qing ( T # ) DING Xiaqi ( T M ) DONG Guangchang (St3fe^) * DONG Yuxin (^SfjfHJf) FAN Engui (fEJSw) * FU Jixiang ( & * # ) FUJITA Hiroshi * GU Chaohao (#j@£E) GUO Benyu ( f ^ H f ) * GUO Zhurui (USttJjg) HE Longguang (©#;&) HONG Anxiang ( $ £ # ) HONG Jiaxing ( « 3 ^ ) * HOUZixin(^SDf) HSIANG Wu-Yi (1%&X) * HUGuoding(iSgfflS)* HU Hesheng ( i ^ M ) HUANG X u a n g u o ( H M ) Jl Min ( £ & ) * JIANG Boju ( # M ) *
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DIFFERENTIAL GEOMETRV AND RELATED TOPICS The International Conference on Modern Mathematics and the International
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