Fundamental tone estimates for elliptic operators in divergence form and geometric applications

Abstract

We establish a method for giving lower bounds for the fundamental tone of elliptic operators in divergence form in terms of the divergence of vector fields. We then apply this method to the Lr operator associated to immersed hypersurfaces with locally bounded (r + 1)-th mean curvature Hr+1 of the space forms Nn+1(c) of constant sectional curvature c. As a corollary we give lower bounds for the extrinsic radius of closed hypersurfaces of Nn+1(c) with Hr+1 > 0 in terms of the r -th and (r + 1)-th mean curvatures. Finally we observe that bounds for the Laplace eigenvalues essentially bound the eigenvalues of a self-adjoint elliptic differential operator in divergence form. This allows us to show that Cheeger’s constant gives a lower bounds for the first nonzero Lr - eigenvalue of a closed hypersurface of Nn+1(c).