Poincar\'e type inequality for hypersurfaces and rigidity results

TL;DR

This paper establishes a general Poincaré type inequality for hypersurfaces under curvature constraints and applies it to derive isoperimetric, rigidity, flatness, and non-existence results for various geometric structures and flows.

Contribution

It introduces a new divergence formula-based approach to derive inequalities and rigidity results for hypersurfaces in space forms and Einstein manifolds.

Findings

Derived isoperimetric inequalities for hypersurfaces.

Proved rigidity results for complete r-minimal hypersurfaces.

Established flatness and non-existence results for certain curvature flows.

Abstract

In this article, under mild constraints on the sectional curvature, we exploit a divergence formula for symmetric endomorphisms to deduce a general Poincar\'e type inequality. We apply such inequality to higher-order mean curvature of hypersurfaces of 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 flatness and non-existence results for self-similar solutions to a large class of fully nonlinear curvature flows.