On the first eigenvalue of the linearized operator of the higher order mean curvature for closed hypersurfaces in space forms

Abstract

n this paper we derive sharp upper bounds for the first positive eigenvalue of the linearized operator of the higher order mean curvature of a closed hypersurface immersed into a Riemannian space form. Our bounds are extrinsic in the sense that they are given in terms of the higher order mean curvatures and the center(s) of gravity of the hypersurface, and they extend previous bounds recently given by Veeravalli [Ve] for the first positive eigenvalue of the Laplacian operator.