Quantitative stability for hypersurfaces with almost constant curvature in space forms. (English) Zbl 1471.53054
Summary: The Alexandrov Soap Bubble Theorem asserts that the distance spheres are the only embedded closed connected hypersurfaces in space forms having constant mean curvature. The theorem can be extended to more general functions of the principal curvatures \(f(k_1,\ldots ,k_{n-1})\) satisfying suitable conditions. In this paper, we give sharp quantitative estimates of proximity to a single sphere for Alexandrov Soap Bubble Theorem in space forms when the curvature operator \(f\) is close to a constant. Under an assumption that prevents bubbling, the proximity to a single sphere is optimally quantified in terms of the oscillation of the curvature function \(f\). Our approach provides a unified picture of quantitative studies of the method of moving planes in space forms.
MSC:
| 53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |
| 53C30 | Differential geometry of homogeneous manifolds |
| 35B50 | Maximum principles in context of PDEs |
| 53C24 | Rigidity results |
Keywords:
space forms geometry; method of the moving planes; Alexandrov soap bubble theorem; quantitative stability; mean curvature; pinchingReferences:
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