Quantitative stability for anisotropic nearly umbilical hypersurfaces. (English) Zbl 1421.53009
Summary: We prove qualitative and quantitative stability of the following rigidity theorem: the only anisotropic totally umbilical closed hypersurface is the Wulff shape. Consider \(n \ge 2\), \(p\in (1, \, +\infty )\) and \(\Sigma \) an \(n\)-dimensional, closed hypersurface in \(\mathbb{R}^{n+1}\), which is the boundary of a convex, open set. We show that if the \(L^p\)-norm of the trace-free part of the anisotropic second fundamental form is small, then \(\Sigma \) must be \(W^{2, \, p}\)-close to the Wulff shape, with a quantitative estimate.
MSC:
| 53A07 | Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces |
| 53C24 | Rigidity results |
| 49Q10 | Optimization of shapes other than minimal surfaces |
| 53A10 | Minimal surfaces in differential geometry, surfaces with prescribed mean curvature |
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