Poincar\'e and mean value inequalities for hypersurfaces in Riemannian manifolds and applications

TL;DR

This paper establishes new Poincaré and mean value inequalities for hypersurfaces in Riemannian manifolds, providing explicit constants and applications to isoperimetric problems, volume estimates, and curvature monotonicity.

Contribution

It introduces novel Poincaré and mean value inequalities with explicit constants for hypersurfaces in Riemannian manifolds, with applications to geometric inequalities and curvature analysis.

Findings

Derived new Poincaré inequalities with explicit constants.

Established isoperimetric inequalities for hypersurfaces.

Provided volume estimates for domains enclosed by self-shrinkers.

Abstract

In the first part of this paper we prove some new Poincar\'e inequalities, with explicit constants, for domains of any hypersurface of a Riemannian manifold with sectional curvatures bounded from above. This inequalities involve the first and the second symmetric functions of the eigenvalues of the second fundamental form of such hypersurface. 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. In the second part of the paper we prove some mean value inequalities and as consequences we derive some monotonicity results involving the integral of the mean curvature.