EXTRINSIC UPPER BOUNDS FOR THE FIRST EIGENVALUE OF ELLIPTIC OPERATORS

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