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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
DOI: https://doi.org/10.1090/S0002-9939-97-04078-1
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.


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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
Email: xiacy@impa.br

DOI: https://doi.org/10.1090/S0002-9939-97-04078-1
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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