A note on starshaped hypersurfaces with almost constant mean curvature in space forms

TL;DR

This paper demonstrates that starshaped hypersurfaces in space forms with nearly constant mean or higher order mean curvature are close to geodesic spheres, highlighting stability properties in geometric analysis.

Contribution

It establishes a stability result for starshaped hypersurfaces with almost constant curvature in space forms, extending previous rigidity theorems.

Findings

Hypersurfaces with nearly constant mean curvature are close to geodesic spheres.

The results apply to higher order mean curvatures as well.

Provides quantitative estimates of closeness to spheres.

Abstract

We show that closed starshaped hypersurfaces of space forms with almost constant mean curvature or almost constant higher order mean curvature are closed to geodesic spheres.