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

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 $p\le n$, 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.

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Cited by in corpus (6)

• Stability from rigidity via umbilicity
• Stability for Serrin's problem and Alexandroff's theorem in warped product manifolds
• Rigidity and quantitative stability for partially overdetermined problems and capillary CMC hypersurfaces
• Stability of the Wulff shape with respect to anisotropic curvature functionals
• The double and triple bubble problem for stationary varifolds: the convex case
• On compact embbeded Weingarten hypersurfaces in warped products