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 tange...

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Bibliographic Details
Published in:International mathematics research notices 2024-09, Vol.2024 (17), p.12111-12135
Main Author: Santilli, Mario
Format: Article
Language:English
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Summary: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.
ISSN:1073-7928
1687-0247
DOI:10.1093/imrn/rnae159