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Rigidity of compact manifolds with boundary and nonnegative Ricci curvature

Author: Changyu Xia
Journal: Proc. Amer. Math. Soc. 125 (1997), 1801-1806
MSC (1991): Primary 53C20, 53C42
MathSciNet review: 1415343
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Abstract: Let $\overline {\Omega }$ be an ($n+1$)-dimensional compact Riemannian manifold with nonnegative Ricci curvature and nonempty boundary $M=\partial \overline {\Omega }$. Assume that the principal curvatures of $M$ are bounded from below by a positive constant $c$. In this paper, we prove that the first nonzero eigenvalue $\lambda _{1}$ of the Laplacian of $M$ acting on functions on $M$ satisfies $\lambda _{1} \geq nc^{2} $ with equality holding if and only if $ \Omega $ is isometric to an $(n+1)$-dimensional Euclidean ball of radius $\frac {1}{c}$. Some related rigidity theorems for $\overline {\Omega }$ are also proved.

References [Enhancements On Off] (What's this?)

  • [BGS] Ballmann, W., Gromov, M. and Schroeder, V., Manifolds of nonpositive curvature, Birkhäuser, Basel-Boston, 1985. MR 87h:53050
  • [CW] Choi, H. I. and Wang, A. N., A first eigenvalue estimate for minimal hypersurfaces, J. Diff. Geom. 18 (1983), 559-562. MR 85d:53028
  • [R] Reilly, R., Applications of the Hessian operator in Riemannian manifolds, Indiana Univ. Math. J. 26 (1977), 459-472. MR 57:13799
  • [Ro1] Ros, A., Compact hypersurfaces with constant higher order mean curvature, Revista matemática Iberroamericana 3 (1987), 447-453. MR 90c:53160
  • [Ro2] Ros, A., Compact hypersurfaces with constant scalar curvature and a congruence theorem, J. Diff. Geom. 27 (1988), 215-220. MR 89b:53096
  • [SS] Schroeder, V. and Strake, M., Rigidity of convex domains in manifolds with nonnegative Ricci and sectional curvature, Comment. Math. Helvetici 64 (1989), 173-186. MR 90h:53042
  • [SZ] Schroeder, V. and Ziller, W., Local rigidity of symmetric spaces, Trans. of the Amer. Math. Soc. 320 (1990), 145-160. MR 90k:53089
  • [X] Xia, C. Y., Rigidity and sphere theorem for manifolds with positive Ricci curvature, manuscripta math. 85 (1994), 79-87. MR 95j:53057
  • [YY] Yang, P. and Yau, S. T., Eigenvalues of the Laplacian of compact Riemannian surfaces and minimal submanifolds, Ann. Scuola norm. Sup. Pisa 7 (1980), 55-63. MR 81m:58084

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Additional Information

Changyu Xia
Affiliation: Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China
Address at time of publication: Instituto de Matematica Pure e Aplicada, Estrada Dona Castorina 110, Jardim Botanico 22460-320, Rio de Janeiro, RJ Brasil

Keywords: Rigidity, manifolds, Ricci curvature
Received by editor(s): December 7, 1995
Additional Notes: This work was supported by the Natural Science Foundation of China, TIT and CNPq.
Communicated by: Christopher Croke
Article copyright: © Copyright 1997 American Mathematical Society

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