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

TL;DR

The paper characterizes large isoperimetric regions and soap bubbles in products of closed manifolds with Euclidean space, revealing geometric structures and stability properties depending on curvature conditions.

Contribution

It proves that large isoperimetric regions in product manifolds are products with Euclidean balls or spheres, depending on curvature, and provides counterexamples with negative curvature.

Findings

Large isoperimetric regions are products with Euclidean balls.

In non-negative Ricci curvature, soap bubbles are products with Euclidean spheres.

Existence of stable soap bubbles with complex projections in negatively curved surfaces.

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.