Extrinsic upper bound for the first eigenvalue of elliptic operators defined on submanifolds and applications

We consider operators defined on a Riemannian manifold M-m by L-T(u) = -div(T del u), where T is a positive definite (1, 1)-tensor such that div(T) = 0. We give upper bounds for the first nonzero eigenvalue lambda(1,T) of L-T in terms of the second fundamental form of an immersion phi of M-m in a manifold with bounded sectional curvature. We apply these results to a particular family of operators operators defined on hypersurfaces of space forms and we prove a stability result. (C) 2000 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.