Rigidity of compact manifolds with boundary and nonnegative Ricci curvature
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- by Changyu Xia PDF
- Proc. Amer. Math. Soc. 125 (1997), 1801-1806 Request permission
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
- Werner Ballmann, Mikhael Gromov, and Viktor Schroeder, Manifolds of nonpositive curvature, Progress in Mathematics, vol. 61, Birkhäuser Boston, Inc., Boston, MA, 1985. MR 823981, DOI 10.1007/978-1-4684-9159-3
- Hyeong In Choi and Ai Nung Wang, A first eigenvalue estimate for minimal hypersurfaces, J. Differential Geom. 18 (1983), no. 3, 559–562. MR 723817
- Robert C. Reilly, Applications of the Hessian operator in a Riemannian manifold, Indiana Univ. Math. J. 26 (1977), no. 3, 459–472. MR 474149, DOI 10.1512/iumj.1977.26.26036
- Antonio Ros, Compact hypersurfaces with constant higher order mean curvatures, Rev. Mat. Iberoamericana 3 (1987), no. 3-4, 447–453. MR 996826, DOI 10.4171/RMI/58
- Antonio Ros, Compact hypersurfaces with constant scalar curvature and a congruence theorem, J. Differential Geom. 27 (1988), no. 2, 215–223. With an appendix by Nicholas J. Korevaar. MR 925120
- Viktor Schroeder and Martin Strake, Rigidity of convex domains in manifolds with nonnegative Ricci and sectional curvature, Comment. Math. Helv. 64 (1989), no. 2, 173–186. MR 997359, DOI 10.1007/BF02564668
- V. Schroeder and W. Ziller, Local rigidity of symmetric spaces, Trans. Amer. Math. Soc. 320 (1990), no. 1, 145–160. MR 958901, DOI 10.1090/S0002-9947-1990-0958901-X
- Chang Yu Xia, Rigidity and sphere theorem for manifolds with positive Ricci curvature, Manuscripta Math. 85 (1994), no. 1, 79–87. MR 1299049, DOI 10.1007/BF02568185
- Paul C. Yang and Shing Tung Yau, Eigenvalues of the Laplacian of compact Riemann surfaces and minimal submanifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 7 (1980), no. 1, 55–63. MR 577325
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
- Email: xiacy@impa.br
- 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
- © Copyright 1997 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 125 (1997), 1801-1806
- MSC (1991): Primary 53C20, 53C42
- DOI: https://doi.org/10.1090/S0002-9939-97-04078-1
- MathSciNet review: 1415343