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

created by derosa on 26 Mar 2018
modified on 13 Dec 2018

[BibTeX]

Preprint

Inserted: 26 mar 2018
Last Updated: 13 dec 2018

Year: 2018

ArXiv: 1803.09118 PDF

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.