On the entropies of hypersurfaces with bounded mean curvature

Abstract

We are interested in the impact of entropies on the geometry of a hypersurface of a Riemannian manifold. In particular, we will be able to compare the volume entropy of a hypersurface with that of the ambient manifold, provided some geometric assumptions are satisfied. This comparison will follow from the existence of an embedded tube around such a hypersurface. Among the consequences of our study of entropies, we point out some new answers to a question of do Carmo on stable Euclidean hypersurfaces of constant mean curvature.