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arXiv · 2203.10334

Poincar\'e type inequality for hypersurfaces and rigidity results

Hilário Alencar·Márcio Batista·Gregório Silva Neto
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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.

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Hilário Alencar, Márcio Batista, Gregório Silva Neto. 2022-03-19. Poincar\'e type inequality for hypersurfaces and rigidity results. https://arxiv.org/abs/2203.10334

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