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

Abstract

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

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References in corpus (1)

• Large isoperimetric regions in the product of a compact manifold with Euclidean space

Cited by in corpus (1)

• Large isoperimetric regions in the product of a compact manifold with Euclidean space