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

Author:

Mario Santilli

Keyword:

Mathematics, Differential Geometry, Differential Geometry (math.DG), Analysis of PDEs (math.AP)

journal:

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date:

2023-08-21 16:00:00

Abstract

Given $ n \geq 2 $ and $ k \in \{2, \ldots , n\} $, we study the asymptotic behaviour of sequences of bounded $C^2$-domains of finite total curvature in $ \mathbb{R}^{n+1} $ converging in volume and perimeter, and with the $ k $-th mean curvature functions converging in $ L^1 $ to a constant. Under natural mean convexity hypothesis, and assuming an $ L^\infty $-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.