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Extrinsic upper bounds for the first eigenvalue of elliptic operators

We consider operators defined on a Riemannian manifold M m by L T (u) = -div(T*u) where T is a positive definite symmetric (1, 1)-tensor such that div(T) = 0. We give an upper bound for the first nonzero eigenvalue λ 1 , T of L T in terms of the second fundamental form of an immersion Φ of Μ m into a Riemannian manifold of sectional curvature bounded above by δ. We apply these results to a particular family of operators defined on hypersurfaces of the space forms and we prove a stability result.