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