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

Authors:	Qing-Ming Cheng

Institution:	Department of Mathematics, Faculty of Science and Engineering, Saga University, Saga 840-8502, Japan

Abstract:	Let be an -dimensional compact hypersurface with constant scalar curvature , , 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 . In this paper, we first study the eigenvalue of the Jacobi operator of . We derive an optimal upper bound for the first eigenvalue of , and this bound is attained if and only if is a totally umbilical and non-totally geodesic hypersurface or is a Riemannian product $S^m(c)\times S^{n-m}(\sqrt{1-c^2})$ , $1\leq m\leq n-1$ .

Keywords:	Hypersurface with constant scalar curvature Jacobi operator mean curvature first eigenvalue and principal curvatures

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