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

We prove that, for every closed (not necessarily convex) hypersurface ς in ℝn+1{mathbb{R} {n+1}} and every p>n{p>n}, the Lp{L {p}}-norm of the trace-free part of the anisotropic second fundamental form controls from above the W2,p{W {2,p}}-closeness of ς to the Wulff shape. In the isotropic setting, we provide a simpler proof. This result is sharp since in the subcritical regime p≤n{pleq 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.