On the Entropies of Hypersurfaces with bounded mean curvature

TL;DR

This paper investigates how entropy measures influence the geometry of hypersurfaces with bounded mean curvature, establishing volume entropy comparisons and addressing a question on stable Euclidean hypersurfaces.

Contribution

It introduces a method to compare hypersurface and ambient manifold entropies under geometric conditions, providing new insights into stable constant mean curvature hypersurfaces.

Findings

Volume entropy of hypersurfaces can be bounded by that of the ambient manifold.

Existence of embedded tubes is crucial for entropy comparison.

New results on stable Euclidean hypersurfaces of constant mean curvature.

Abstract

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