Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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?)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 53C20, 53C42

Retrieve articles in all journals with MSC (1991): 53C20, 53C42

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