Finite Total Curvature and Soap Bubbles With Almost Constant Higher-Order Mean Curvature

Abstract

Given $ n \geq 2 $ and $ k \in \{2, \ldots , n\} $, we study the asymptotic behaviour of sequences of bounded $C^{2}$-domains, whose $ k $-th mean curvature functions converge in $ L^{1} $-norm to a constant. Under certain curvature assumptions, we prove that finite unions of mutually tangent balls are the only possible limits with respect to convergence in volume and perimeter. The key novelty of our statement lies in the fact that we do not assume bounds on the exterior or interior touching balls.