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

TL;DR

This paper investigates the asymptotic behavior of sequences of bounded domains with finite total curvature in Euclidean space, showing that under certain conditions, the limits are unions of tangent balls, advancing understanding of soap bubble-like structures.

Contribution

It proves the uniqueness of limit shapes as unions of tangent balls for sequences of domains with converging mean curvature, without requiring uniform bounds on touching balls.

Findings

Limits are unions of tangent balls under given conditions.

First result without uniform touching ball bounds.

Advances understanding of geometric limits in curvature problems.

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.