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hal-00145766, version 1

## EXTRINSIC UPPER BOUNDS FOR THE FIRST EIGENVALUE OF ELLIPTIC OPERATORS

Jean-Francois Grosjean () 1

Hokkaido Math. J. 33, 2 (2004) 319-339

Abstract: We consider operators defined on a Riemannian manifold $M^m$ by $\lt(u)=-div(T\nabla u)$ where $T$ is a positive definite $(1,1)$-tensor such that $div(T)=0$. We give an upper bound for the first nonzero eigenvalue $\lat$ of $\lt$ in terms of the second fundamental form of an immersion $\phi$ of $M^m$ into a Riemannian manifold of bounded sectional curvature. We apply these results to a particular family of operators defined on hypersurfaces of space forms and we prove a stability result.

• 1:  Institut Elie Cartan Nancy (IECN)
• CNRS : UMR7502 – INRIA – Université Henri Poincaré - Nancy I – Université Nancy II – Institut National Polytechnique de Lorraine (INPL)
• Domain : Mathematics/Differential Geometry
• Keywords : r-th mean curvature – Reilly's inequality

• hal-00145766, version 1
• oai:hal.archives-ouvertes.fr:hal-00145766
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• Submitted on: Friday, 11 May 2007 15:00:03
• Updated on: Thursday, 24 May 2007 14:59:23