New Poincaré inequalities for mean and scalar curvatures and applications

Hilário Alencar, Gregório Silva Neto

Published 2015-02-05Version 1

In this paper we prove some new Poincar\'e inequalities, with explicit constants, for domains of any hypersurfaces of Euclidean space, hyperbolic space or Euclidean sphere, involving the mean and the scalar curvature of such hypersurfaces. We apply these inequalities to derive some isoperimetric inequalities and to estimate the volume of domains enclosed by compact self-shrinkers in terms of its scalar curvature.