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 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.

Rigidity theorems for minimal submanifolds in a hyperbolic space

“…Remark 1.5. An interesting problem suggested by the referee is to understand the relation between the assumption on the total curvature and the volume entropy of the hypersurface M. For some ideas, one can see the pioneer articles [6,7] and the more recent [24] and references therein. In particular [24, Theorem 6.1].…”

A survey about a Do Carmo’s question on complete stable constant mean curvature hypersurfaces

“…For a long time, the only concrete answers to Do Carmo's question were obtained by taking assumptions on total curvature. In 2016 [28], S. Ilias, the author and M. Soret considered a different point of view and study Do Carmo's question with respect to assumptions on the volume growth.…”

A Note on Constant Mean Curvature Foliations of Noncompact Riemannian Manifolds

“…By [12] ( eorem 7.6), a manifold L is parabolic if and only if there exists a smooth exhaustion function v on L such that…”

“…Later, the answer was proved to be positive for n = 3 and 4 by Elbert et al [10] and independently by Cheng [11], using a Bonnet-Myers' type method. Moreover, the authors gave a positive answer to do Carmo's question in the following cases: (1) a hypersurface with zero volume entropyof a space form of any dimension [12] (Corollary 8). (2) a hypersurface of R n+1 , H n+1 , n ≤ 5 with total curvature with polynomial growth [13] (Corollary 6.3).…”