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

In the first part of this paper we prove some new Poincar\'e inequalities, with explicit constants, for domains of any hypersurfaces of Riemannian manifolds with sectional curvature bounded from above, involving the first and the second symmetric functions of the eigenvalues of the second fundamental form 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. 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.