Soap bubbles and isoperimetric regions in the product of a closed manifold with Euclidean space

2013/12/21 by Jesús Gonzalo Pérez, Pérez, Jesús Gonzalo · 1 citation
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.1312.6311

openalex publication_date 2013/12/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/02

Abstract

For any closed Riemannian manifold X we prove that large isoperimetric regions in X×\mathbb Rⁿ are of the form X×(Euclidean ball). We prove that if X has non-negative Ricci curvature then the only soap bubbles enclosing a large volume are the products X×(Euclidean sphere). We give an example of a surface X, with Gaussian curvature negative somewhere, such that the product X×\mathbb R contains stable soap bubbles of arbitrarily large enclosed volume which do not even project surjectively onto the X factor.

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