Global Differential Geometry and Global Analysis: Proceedings, Berlin 1991

288 Pages·1991·11.14 MB·English

by  D. Ferus

Lecture Notes in Mathematics 1481
Editors:
A. Dold, Heidelberg
B. Eckmann, Zttrich
.F Takens, Groningen
D. Ferns U. Pinkall
U. Simon B. Wegner ).sdE(
Global Differential
Geometry dna
Global Analysis
Proceedings of a Conference held in
Berlin, 15-20 June, 1990
galreV-regnirpS
Berlin Heidelberg NewYork
London Paris Tokyo
Kong Hong Barcelona
tsepaduB
Editors
Dirk Fetus
Uh'ich Pinkall
Udo Simon
Berd Wegner
Fachbereich Mathematik
Technische Universit~it Berlin
W-1000 Berlin ,21 FRG
Mathematics Subject Classification (1991): 53-06, 58-06, 53A04, 53A07, 53A10,
53A15, 53A60, 53C30, 53C42, 58A50, 58D05, 58E05, 58G25.58G30
ISBN 3-540-54728-2 Springer-Verlag Berlin Heidelberg New York
ISBN 0-387-54728-2 Springer-Verlag New York Berlin Heidelberg
This work is subject to copyright. All rights are reserved, whether the whole or part of
the material is concerned, specifically the rights of translation, reprinting, re-use of
illustrations, recitation, broadcasting, reproduction on microfilms or in any other way,
and storage in data banks. Duplication of this publication or parts thereof is permitted
only under the provisions of the German Copyright Law of September 9, I965, in its
current version, and permission for use must always be obtained from Springer-Verlag.
Violations are liable for prosecution under the German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1991
Printed in Germany
Typesetting: Camera ready by author
Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr.
46/3140-543210 - Printed on acid-free paper
Introduction
This conference continued a long tradition of similar meetings held at TU Berlin in earlier
years. It was, however, for two reasons very distinct from its predecessors:
For the first time the EC-Project "Global Analysis, Geometry and Applications" provided
the framework, bringing together more than fifty representatives of the twelve
participating institutes from all over Western Europe. The exchange between our project
and the recent developements in geometry and analysis outside the realms of the EC
program was a major objective during the meeting.
The second novelty of this conference was caused by the unexpected political changes in
Berlin, Germany and East Europe. Encounters with mathematicians from eastern
countries have always been a special feature of the Berlin conferences, but this time there
came more than sixty mathematicians from the former DDR and other socialist countries.
Many of us had know each other for a long time through our publications, but had never
met personally. It became quite obvious, that the conference laid the foundation for a
number of future east-west cooperations.
The total number of more than 200 participants was twice that of earlier years. About 40
colleagues came from overseas.
The notes presented in this volume give only an incomplete selection of the topics covered
during the conference. We therefore include a list of the participants and of all invited
titles.
We are grateful for the support that made this conference possible. Our thanks go to
European Community (Contract SC 1-0039-C AM)
Deutsche Forschungsgemeinschaft
Deutscher Akademischer Austauschdienst
Senat yon Berlin
Technische Universitat Berlin.
We also thank Springer Verlag for the publication of a third proceedings volume on a
Global Geometry and Analysis Conference at the TU Berlin.
BerndW egner, Dirk Ferus, Ulrich Pinkall, Udo Simon
Contents
E. Belchev, S. Hineva: 1
On the minimal hypersurfaces of a locally symmetric manifold.
N. BlasiS, N. Bokan, P.Gilkey: 5
The spectral geometry of the Laplacian and the conformal Laplacian for manifolds with
boundary.
J. Bolton, W. M. Oxbury, L. Vrancken, L. M. Woodward: 18
Minimal immersions of RP 2 into CP n.
W. Cie~lak, A. Miernowski, W. Mozgawa: 28
Isoptics of a strictly convex curve.
F. Dillen, L. Vrancken: 36
Generalized Cayley surfaces.
A. FerrAndez, O. J. Garay, P. Lucas: 48
On a certain class of conformally fiat Euclidean hypersurfaces.
P. Gauduchon: 55
Self-dual manifolds with non-negative Ricci operator.
B. Hajduk: 62
On the obstruction group to existence of Riemannian metrics of positive scalar curva-
ture.
U. Hamenstiidt: 74
Compact manifolds with ¼-pinched negative curvature.
J. Jost~ Xiao-Wei Peng: 80
The geometry of moduli spaces of stable vector bundles over Riemannian surfaces.
O. Kowalski, F. Tricerri: 98
A canonical connection for locally homogeneous Riemannian manifolds.
M. Kozlowski: 105
Some improper affine spheres in Aa.
R. Kusner: 109
A maximum principle at infinity and the topology of complete embedded surfaces with
constant mean curvature.
IIIV
Li An-Min: 611
AJ~ne completeness and Euclidean completeness.
U. Lumiste: 721
On submanifolds with parallel higher order fundamental form in Euclidean spaces.
A. Martlnez~ t F. Mil~in: 931
Convex aj~ne surfaces with constant a~ne mean curvature.
M. Min-Oo~ E. A. Ruh~ P. Tondeur: 641
Transversal curvature and tautness for Riemannian foliations.
S. Montiel~ A. Ros: 841
SchrSdinger operators associated to a holomorphic map.
oD Motreanu: 671
Generic existence of Morse functions on infinite dimensional Riemannian manifolds
and applications.
B. Opozda: 681
Some extensions of Radon's theorem.
H.-B. Rademacher: 391
Generalized Killing spinors with imaginary Killing function and conformal Killing fields.
V. Rosenhaus: 002
On prolongation and invariance algebras in superspace.
T. Sasaki: 112
On the Veronese embedding and related syatem of differential equations.
R. Schimming: 942
Generalizations of harmonic manifolds.
R. Schmid: 952
Diffeomorphism groups, pseudodifferential operators and r-matrices.
A. M. Shelekhov: 562
On the theory of G-webs and G-loops.
Wang Changping: 272
Some examples of complete hyperbolic aj~ne 2-spheres in 3 R
List of Participants 282
On the minimal hypersurfaces of a locally symmetric manifold
Stefana Hineva, Evgeni Belchev
1. Introduction.
Let M'* be an n-dimensional hypersurface which is minimally immersed in an n+l-
dimensional locally symmetric manifold N n+l. Let h be the second fundamental form
of this immersion. We denote by S the squaxe of the length of h. In this paper we give
conditions in which S is constant and show when S vanishes, i.e. when M n is totally
geodesic.
2. Local formulas for a minimal hypersurface.
In this section we shall compute the Laplacian of the second fundamental form
of a minimal hypersurface M ~' of a locally symmetric space N "+1 . We shall follow
closely the exposition in [1], even we shall take some formulas directly from [1]. Let
el,e2~... ,en~en+ 1 be a local frame of orthonormal vector fields in N 1+= such that,
restricted to M n the vectors el, e2,..., en are tangent to M'~; the vector en+l is normal
to M n. We shall make use of the following convention on the ranges of indices: 1 <_
i,j, k,... < n, 1 < A, B, C,... < n + 1 and we shall agree that repeated indices are
summed over the respective ranges.
With respect to the frame el, e2,..., e,, e,,+l let us denote by hij the components
of the second fundamental form of ~,r~3 and by k Ri'~ and KDBc the components of the
curvature tensors of M n and N "+1 respectively. We call
H = 1_ ~ hiien+l (2.1)
2?
i=1
the mean curvature vector of M n. The square of the length of the second fundamental
form of M" is given by
s = (h j) (2.2)
i,j=l
S and H 2 -1 --- ~-'r(~i=l n hii) ~ are independent of the choice of the orthonormal frame.
It is well known that for an arbitrary submanifold ~ .~r of a Riemannian manifold
N ~+1 we have
Rijkl "~" Kijkl + hlkh.jt - hithjk (Gauss equation of Mn), (2.3)
hijk -- hikj = Kink+ I = --I( .... ij +J k (Codazzi equation of Mn). (2.4)
Here hijk is the covariant derivative of hij.
(2.5)
hijkl -- hijlk : hi,,Rjmkl + hrnjRin~.l •
This equation is obtained from (2.15) in [1], when a --- p = n + 1. Here hljkl is the
covariant derivative of hijk.
Kn..t-1 ijk;l = El.k-inj+kll -n+l-- f('i n+l khJ l -- .l.t..i..j +1 n+l hkl + I('ijk.-hrnm l" (2.6)
This equation is obtained from (2.17) in [1]. Here r~xij,k;t 'n+l is the restriction to M" of the
E;DCB'(f
covariant derivative *A Of I(AcD as a curvature tensor of N n+ l.
We consider k r.t~.ij ~'n+l as a section of the bundle T ± ® T* ® T* ® T* where T is the
tangent bundle T = T(M), T* = T*(M) - the cotangent bundle and T ± = T±(M)
- the normal bundle. *'iTjcktn +I is the covariant derivative of f{ijk! n+l with respect to the
covariant differentiation which maps a section of T ± ® T* ® T* ® T* into a section of
T -t- @ T* ® T* ® T* ® T*. Kg~ must be distinguished from K~+;~
Since we suppose that N n+l is a locally symmetric one, then KAcD;E = 0 and
from (2.6) we obtain that
gn+l r(n+l .... +1 h,., "'~ (2.7)
ijkl = "in+l khjl + lxijn+l ~'-- I{ijkhrnt"
The Laplacian Ahij of the second fundamental form h of M n is defined by
tl
~hlj = ~ ]Zijkk. (2.g)
k=l
From (2.4) we obtain
r,"n+l Kn+l
Ahij = hikjk -- ltijkk = hkijk -- ijkk" (2.9)
From (2.5) we have
hkljk = hkikj + hkmRi"jk + hmil:tk'~k. (2.10)
In (2.10) replacing hklkj by hkkij -- K~. +] from (2.4) and putting it in (2.9) we obtain
..n+l T.n+1 m m (2.11)
Ahij = hkkij .lkkikj .lkijkk + hkmRij k + hmiRkj k.
-- --
Then from (2.11), (2.7) and (2.3) it follows that
Ahij : hkkij -- hkkKij ,n+l _ hiji(~.+~, + k (2.12)
n+l
ar m m 2
hmjRkik + hmiJr~kjk + 2hmkI'Cij k hmil~mihkk + hijhmk.
--
1 ~r l+neii'h
As M n is minimal, i.e. H = ~ ~i=1 = 0, then from (2.12) we obtain for the
Ahij
Laplacian of the second fundamental form of a minimal hypersurface M ~ of N "+1
2 (2.13)
Ahij = -hijack . . . . +,1+ 1 k + hmjRkik m + h,niRkjk rn + 2h,,kRO'k) m + hijhmk.
3. Minimal hypersurfaces of a special class of a locally symmetric mani-
folds.
In this section we assume that the ambient space N "+1 is a locally symmetric one
with sectional curvature K2v~ satisfying the condition < 6 I(N= <_ 1 at all points x 6 M ~'
1
and 6 > 7" We shall prove the following theorem:
Theorem 1. Let M" be a compact minimal hypersurface of a locally symmetric
manifold N n+l whose sectional curvature KN~, at 1M points x 6 M ~' satisfies
1
6 < KN~ _< 1 for 6 > ~. (3.1)
If the square of the length S of the second fundamental form of M n satisfies the condition
S < (26 - 1)n (3.2)
n-1
then S is constant.
Proof. For the Laplacian AS of S = ~i:j=l(hlj) 2 we have
1AS= ~ hijAhij + ~ (Vhij) .2 (3.3)
i,j= l i,j,k= l
We replace in (3.3) ~hij from (2.13) and obtain
1AS > -Lnij ,, ~) 2r~-n+llxkn+lk+2(hijhmjR~.ik+hijhmk.Rijk)+(hij)m 2 2 h,.,k. (3.4)
We shall prove that the right side of (3.4) is non-negative. If we denote by Ii the
eigenvalues of the matrix (hij) of the second from of n, M then Yau's formula (10.9)
from ]2[ gives that
n
2(ho.hmj/{k"~k + hijhmkI{in]k) = E(,~i - ,~j)2Rijij (3.5)
ij
II
ij
where K(x) is a function which assigns to each point x 6 M n the infinimum of the
sectional curvature of M n at that point.
For the sectional curvature KM(a) of an arbitrary submanifold M n of a ttiema~nian
manifold N n+p at a point x E M n for the plane a we have from ]4[ the following estimate:
1 n2H 2
KM(a) > KN(~) + 2(n-T )S )6.3(
where KN(a) is the sectional curvature of N p+n at the point z 6 M". Because of (3.1)
and H = 0 the inequality (3.6) takes the form
1 (3.7)
Taking into account (3.7) for the left side of (3.5) we obtain the following lower estimate:
2( kimkRjmhjih + hijhmk kjniR ) _> 2n6S - nS .2 (3.8)
Because of (3.1) we have
r,"~+ 1 < n. (3.9)
gx k n+l k
For AS we obtain from (3.4) because of (3.8)7 (3.9) and (2.2) that
JS - >_ )i .]s)1 (3.10)
- - -
From (3.10) in view of (3.2) we obtain that
IAS >_ .O (3.11)
2
Next, from the Hopf principle it follows that S = const. Hence the theorem is proved.
Theorem 2. Let M n be a complete, connected minimal hypersurface of a locally
symmetric manifold N "+1 whose sectional curvature KN. at all points x E M" satisfies
(3.1). If the square of the length S of the second fundamental form of M n satisfies
S < 26 - 1, (3.12)
then M n is totally geodesic.
Proof. From (3.10) and (3.12) it follows that
1
-~AS > S(26 - 1). (3.13)
Since S is bounder above and the sectional curvature of M n is bounded below, we claim
that S -- 0 everywhere on M n. In fact, if for some point p E M ~' we had S(p) = a > ,O
then from (3.13) we should have ½AS _> a(26 - 1) = const. > 0, and thus for all points
q E M" for which S(q) > S(p) we ought have ½AS(q) > a(26- 1) > 0 which contradicts
Omori's theorem A' in [3]. So, S = 0 everywhere on M n which means that M n is totally
geodesic. Hence the theorem is proved.
References.
]1[ Chern, S.S., M. do Carmo and Kobayashi S.: Minimal submanifolds of a sphere
with second fundamental form of constant length. Functional Analysis and Related
Fields, 59-75 (1970).
]2[ Yau, S. T.: Submanifolds with constant mean curvature II. Amer. J. of Math. 97,
No. 1, 76-100 (1975).
]3[ Omori, H.: Isometric immersions of 1-<iemannian manifolds. J. Math. Soc. Japan
19,205-214 (1967).
]4[ Hineva, S. T.: Submanifolds and the second fundamental tensor. Lecture Notes in
Math. 1156, 194-203 (1984).
This paper is in final form and no version will appear elsewhere.
Faculty of Mathematics and Informatics
University of Sofia
Anton Ivanov Str. 5
1126 Sofia, Bulgaria