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An inequality of “Reilly”-type for submanifolds of the hyperbolic space. (Une inégalité du type ”Reilly” pour les sous-variétés de l’espace hyperbolique.) (French) Zbl 0758.53029
Given a compact submanifold in the Euclidean space, \(M^ n\subset \mathbb{R}^ m\), R. C. Reilly [ibid. 52, 525-533 (1977; Zbl 0382.53038)] proved the inequality \(n\int_ M H^ 2\geq \lambda_ 1 V\), \(\lambda_ 1\) and \(V\) being the first eigenvalue of the Laplacian and the volume of \(M\), and \(H\) being the mean curvature vector of the immersion. If the equality holds, then \(M\) is minimally contained in an umbilical hypersurface of \(\mathbb{R}^ m\). The authors obtain an analogous result for submanifolds of the hyperbolic space. In this case they prove the inequality \(n\int_ M H^ 2\geq(\lambda_ 1+n)V\). Finally, the authors give applications to an eigenvalue estimate for \(\Delta+W\) on submanifolds and to the study of stability of constant mean curvature hypersurfaces.

53C40 Global submanifolds
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