Weighted Hsiung-Minkowski formulas and rigidity of umbilical hypersurfaces

2016/08/31 by Kwok‐Kun Kwong, Kwong, Kwok-Kun, Hojoo Lee +3
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1608.08955

openalex publication_date 2016/08/31 · openalex created_date 2016/09/16 · openalex updated_date 2026/07/28

Abstract

We use the weighted Hsiung-Minkowski integral formulas and Brendle's inequality to show new rigidity results. First, we prove Alexandrov type results for closed embedded hypersurfaces with radially symmetric higher order mean curvature in a large class of Riemannian warped product manifolds, including the Schwarzschild and Reissner-Nordström spaces, where the Alexandrov reflection principle is not available. Second, we prove that, in Euclidean space, the only closed immersed self-expanding solitons to the weighted generalized inverse curvature flow of codimension one are round hyperspheres.

Citations

• Isoperimetric inequalities for eigenvalues by inverse mean curvature flow
• An integral formula and its applications on sub-static manifolds

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