Stability of hypersurfaces with constant (r+1)-th anisotropic mean curvature

2008/12/01 by Yijun He, Haizhong Li · 3 citations
Mathematics · #Analytic and geometric function theory #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.1215/ijm/1258554364

crossref issued 2008/12/01 · crossref published 2008/12/01 · crossref published-print 2008/12/01 · openalex publication_date 2008/12/01 · crossref created 2019/03/07 · crossref deposited 2024/01/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28 · crossref indexed 2026/07/29

Abstract

Given a positive function F on Sⁿ which satisfies a convexity condition, we define the r-th anisotropic mean curvature function H^F_r for hypersurfaces in ℝⁿ⁺¹ which is a generalization of the usual r-th mean curvature function. Let X : M→ℝⁿ⁺¹ be an n-dimensional closed hypersurface with H^F_r+1=constant, for some r with 0≤ r≤ n-1, which is a critical point for a variational problem. We show that X(M) is stable if and only if X(M) is the Wulff shape.

Cited by

• Symmetrization with respect to mixed volumes
• On the stability of the Wulff shape
• Inverse anisotropic mean curvature flow and a Minkowski type inequality

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