Poincaré type inequality for hypersurfaces and rigidity results

Abstract

In this paper, we deal with general divergence formulas involving symmetric endomorphisms. Using mild constraints in the sectional curvature and such divergence formulas we deduce a very general Poincaré type inequality. We apply such general inequality for higherorder mean curvature, in space forms and Einstein manifolds, to obtain several isoperimetric inequalities, as well as rigidity results for complete r-minimal hypersurfaces satisfying a suitable decay of the second fundamental form at infinity. Furthermore, using these techniques, we prove the flatness and non existence results for self-similar solutions of a large class of fully nonlinear curvature flows.