We prove qualitative and quantitative stability of the following rigidity theorem: the only anisotropic totally umbilical closed hypersurface is the Wulff shape. Consider n≥ 2 , p∈(1,+∞) and Σ an n-dimensional, closed hypersurface in Rn+1, which is the boundary of a convex, open set. We show that if the Lp-norm of the trace-free part of the anisotropic second fundamental form is small, then Σ must be W2,p-close to the Wulff shape, with a quantitative estimate.
Quantitative stability for anisotropic nearly umbilical hypersurfaces
De Rosa, Antonio;
2019
Abstract
We prove qualitative and quantitative stability of the following rigidity theorem: the only anisotropic totally umbilical closed hypersurface is the Wulff shape. Consider n≥ 2 , p∈(1,+∞) and Σ an n-dimensional, closed hypersurface in Rn+1, which is the boundary of a convex, open set. We show that if the Lp-norm of the trace-free part of the anisotropic second fundamental form is small, then Σ must be W2,p-close to the Wulff shape, with a quantitative estimate.
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