Quantitative stability for hypersurfaces with almost constant mean curvature in the hyperbolic space

TL;DR

This paper establishes precise stability estimates for hypersurfaces with nearly constant mean curvature in hyperbolic space, using the method of moving planes, leading to new geometric pinching results.

Contribution

It provides the first quantitative stability analysis for Alexandrov's theorem in hyperbolic space, connecting geometric closeness to curvature and touching ball conditions.

Findings

Quantitative stability estimates for hypersurfaces in hyperbolic space.

New pinching theorem for hypersurfaces with almost constant mean curvature.

Application of the method of moving planes in a quantitative hyperbolic setting.

Abstract

We provide sharp stability estimates for the Alexandrov Soap Bubble Theorem in the hyperbolic space. The closeness to a single sphere is quantified in terms of the dimension, the measure of the hypersurface and the radius of the touching ball condition. As consequence we obtain a new pinching result for hypersurfaces in the hyperbolic space. Our approach is based on the method of moving planes. In this context we carefully review the method and we provide the first quantitative study in the hyperbolic space.