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

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 $L_r$ operator associated to immersed hypersurfaces with locally bounded $(r+1)$-th mean curvature $H_{r+1}$ of the space forms ${\mathbb{N}}^{n+1}(c)$ of curvature $c$. As a corollary we give lower bounds for the extrinsic radius of closed hypersurfaces of ${\mathbb{N}}^{n+1}(c)$ with $H_{r+1}>0$ 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 $L_r$-eigenvalue of a closed hypersurface of ${\mathbb{N}}^{n+1}(c)$.