# Quantitative $W^{2, \, p}$-stability for almost Einstein hypersurfaces

## TL;DR

This paper proves that hypersurfaces nearly satisfying Einstein conditions in an $L^p$ sense are quantitatively close to round spheres in the $W^{2,p}$ norm, extending classical rigidity results with stability estimates.

## Contribution

It provides a quantitative stability version of the classical Einstein hypersurface characterization, establishing $W^{2,p}$-closeness under almost Einstein conditions.

## Key findings

- Hypersurfaces nearly Einstein are close to spheres in $W^{2,p}$ norm.
- The stability estimate is explicit and quantitative.
- The result generalizes classical rigidity theorems to an approximate setting.

## Abstract

It is a well known fact that, if $\Sigma$ is an Einstein hypersurface with positive scalar curvature, then it is a round sphere. We give a stable version of this result showing that if a hypersurface is almost Einstein in a $L^p$-sense, then it is $W^{2, \, p}$ - close to the round sphere. The result is given in a quantitative way.