Absence of bubbling phenomena for non convex anisotropic nearly umbilical and quasi Einstein hypersurfaces

TL;DR

This paper establishes a sharp quantitative relationship between the anisotropic second fundamental form and the closeness of hypersurfaces to the Wulff shape, proving the absence of bubbling phenomena for certain non-convex hypersurfaces.

Contribution

It introduces a new sharp stability estimate controlling hypersurface closeness to Wulff shape via anisotropic second fundamental form in non-convex settings.

Findings

Control of hypersurface shape by anisotropic second fundamental form

Absence of bubbling phenomena in non-convex hypersurfaces for p > n

Improved estimates in convex case

Abstract

We prove that, for every closed (not necessarily convex) hypersurface $\Sigma$ in $\mathbb{R}^{n+1}$ and every $p>n$, the $L^p$-norm of the trace-free part of the anisotropic second fundamental form controls from above the $W^{2,p}$-closeness of $\Sigma$ to the Wulff shape. In the isotropic setting, we provide a simpler proof. This result is sharp since in the subcritical regime $p\leq n$, the lack of convexity assumptions may lead in general to bubbling phenomena. Moreover, we obtain a stability theorem for quasi Einstein (not necessarily convex) hypersurfaces and we improve the quantitative estimates in the convex setting.