Monotonicity formula for complete hypersurfaces in the Hyperbolic space and applications

TL;DR

This paper establishes a monotonicity formula for the mean curvature integral of complete hypersurfaces in hyperbolic space, leading to bounds and divergence results that deepen understanding of their geometric properties.

Contribution

It introduces a new monotonicity formula for mean curvature integrals in hyperbolic space, providing novel bounds and divergence results for complete hypersurfaces.

Findings

Lower bound for the mean curvature integral

Divergence of the mean curvature integral to infinity

Monotonicity formula for hypersurfaces in hyperbolic space

Abstract

In this paper we prove a monotonicity formula for the integral of the mean curvature for complete and proper hypersurfaces of the hyperbolic space and, as consequences, we obtain a lower bound for the integral of the mean curvature and that the integral of the mean curvature is infinity.