Stability of hypersurfaces with constant r-th anisotropic mean curvature

2008/01/23 by Yijun He, Haizhong Li, He, Yijun +1
Mathematics · Physics and Astronomy · #49Q10 (Secondary) #53A10 (Primary) #53C42 #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.0801.3561

openalex publication_date 2008/01/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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