Asian Journal of Mathematics

Volume 21 (2017)

Number 4

Poincaré and mean value inequalities for hypersurfaces in Riemannian manifolds and applications

Pages: 697 – 720

DOI: https://dx.doi.org/10.4310/AJM.2017.v21.n4.a4

Authors

Hilário Alencar (Instituto de Matemática, Universidade Federal de Alagoas, Maceió, AL, Brazil)

Gregório Silva Neto (Instituto de Matemática, Universidade Federal de Alagoas, Maceió, AL, Brazil)

Abstract

In the first part of this paper we prove some new Poincaré 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.

Keywords

Poincaré inequality, isoperimetric inequality, monotonicity, hypersurfaces, mean curvature, scalar curvature

2010 Mathematics Subject Classification

53C21, 53C42

Received 11 October 2015

Published 25 August 2017