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First eigenvalue of a Jacobi operator of hypersurfaces with a constant scalar curvature

Author: Qing-Ming Cheng
Journal: Proc. Amer. Math. Soc. 136 (2008), 3309-3318
MSC (2000): Primary 53C42; Secondary 58J50
Published electronically: May 5, 2008
MathSciNet review: 2407097
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Abstract: Let $ M$ be an $ n$-dimensional compact hypersurface with constant scalar curvature $ n(n-1)r$, $ r> 1$, in a unit sphere $ S^{n+1}(1)$. We know that such hypersurfaces can be characterized as critical points for a variational problem of the integral $ \int_MHdM$ of the mean curvature $ H$. In this paper, we first study the eigenvalue of the Jacobi operator $ J_s$ of $ M$. We derive an optimal upper bound for the first eigenvalue of $ J_s$, and this bound is attained if and only if $ M$ is a totally umbilical and non-totally geodesic hypersurface or $ M$ is a Riemannian product $ S^m(c)\times S^{n-m}(\sqrt{1-c^2})$, $ 1\leq m\leq n-1$.

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Qing-Ming Cheng
Affiliation: Department of Mathematics, Faculty of Science and Engineering, Saga University, Saga 840-8502, Japan

Keywords: Hypersurface with constant scalar curvature, Jacobi operator, mean curvature, first eigenvalue and principal curvatures
Received by editor(s): November 14, 2006
Received by editor(s) in revised form: August 2, 2007
Published electronically: May 5, 2008
Additional Notes: The author’s research was partially supported by a Grant-in-Aid for Scientific Research from JSPS
Communicated by: Richard A. Wentworth
Article copyright: © Copyright 2008 American Mathematical Society