Finite total curvature and soap bubbles with almost constant higher-order mean curvature

2023/08/22 by Mario Santilli, Santilli, Mario · 1 citation
Mathematics · #Analysis of PDEs (math.AP) #Analytic and geometric function theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2308.11253

openalex publication_date 2023/08/22 · openalex created_date 2023/08/24 · openalex updated_date 2026/07/28

Abstract

Given n ≥ 2 and k ∈ \2, … , n\ , we study the asymptotic behaviour of sequences of bounded C²-domains of finite total curvature in ℝⁿ⁺¹ converging in volume and perimeter, and with the k -th mean curvature functions converging in L¹ to a constant. Under natural mean convexity hypothesis, and assuming an L^∞ -control on the mean curvature outside a set of vanishing area, we prove that finite unions of mutually tangent balls are the only possible limits. This is the first result where such a uniqueness is proved without assuming uniform bounds on the exterior or interior touching balls.

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• Hessian operators, overdetermined problems, and higher order mean curvatures: symmetry and stability results

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