# Quantitative stability for anisotropic nearly umbilical hypersurfaces

## TL;DR

This paper establishes both qualitative and quantitative stability results for anisotropic hypersurfaces, showing that nearly umbilical shapes are close to the Wulff shape in a precise mathematical sense.

## Contribution

It provides the first quantitative stability estimate for anisotropic hypersurfaces near the Wulff shape, extending classical rigidity results.

## Key findings

- Small $L^p$ norm of trace-free anisotropic second fundamental form implies closeness to Wulff shape.
- Quantitative estimates are derived for the $W^{2,p}$-closeness.
- Results hold for convex, closed hypersurfaces in $
eal^{n+1}$.

## Abstract

We prove a qualitative and a quantitative stability of the following rigidity theorem: an anisotropic totally umbilical closed hypersurface is the Wulff shape. Consider $n \geq 2$, $p\in (1, \, +\infty)$ and $\Sigma$ an $n$-dimensional, closed hypersurface in $\mathbb{R}^{n+1}$, boundary of a convex, open set. We show that if the $L^p$ norm of the trace-free part of the anisotropic second fundamental form is small, then $\Sigma$ must be $W^{2, \, p}$-close to the Wulff shape, with a quantitative estimate.