# Quantitative stability for hypersurfaces with almost constant curvature in space forms

## TL;DR

This paper provides sharp quantitative estimates for how close hypersurfaces with nearly constant curvature are to spheres in space forms, extending classical results to more general curvature functions.

## Contribution

It introduces a unified approach to quantify the proximity of hypersurfaces to spheres when the curvature function is nearly constant, generalizing the Alexandrov Soap Bubble Theorem.

## Key findings

- Quantitative estimates relate curvature oscillation to proximity to spheres.
- Extension of classical theorems to general curvature functions.
- A unified method for quantitative stability in space forms.

## Abstract

The Alexandrov Soap Bubble Theorem asserts that the distance spheres are the only embedded closed connected hypersurfaces in space forms having constant mean curvature. The theorem can be extended to more general functions of the principal curvatures $f(k_1,\ldots,k_{n-1})$ satisfying suitable conditions. In this paper we give sharp quantitative estimates of proximity to a single sphere for Alexandrov Soap Bubble Theorem in space forms when the curvature operator $f$ is close to a constant. Under an assumption that prevents bubbling, the proximity to a single sphere is quantified in terms of the oscillation of the curvature function $f$. Our approach provides a unified picture of quantitative studies of the method of moving planes in space forms.