None

Peter Roquette, Joachim Cuntz, David Masser, Tobias Colding, Daniel Huybrechts, Jun-Muk Hwang, Jakob Stix, Rainer Weissauer, Geordie Williamson, Franz Stückle

Abstract

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{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.

Locations

• Journal für die reine und angewandte Mathematik (Crelles Journal) - View - PDF