Double Bubbles with High Constant Mean Curvatures in Riemannian Manifolds

TL;DR

This paper proves the existence and multiplicity of double bubble configurations with high constant mean curvature in Riemannian manifolds, extending classical geometric results to more general curved spaces.

Contribution

It introduces new existence and multiplicity results for double bubbles with high constant mean curvature in Riemannian manifolds, using perturbation and variational methods.

Findings

Existence of large constant mean curvature double bubbles in Riemannian manifolds.

Multiplicity results based on Lusternik-Schnirelman theory.

Special cases with symmetric boundary mean curvatures.

Abstract

We obtain existence of double bubbles of large and constant mean curvatures in Riemannian manifolds. These arise as perturbations of geodesic standard double bubbles centered at critical points of the ambient scalar curvature and aligned along eigen-vectors of the ambient Ricci tensor. We also obtain general multiplicity results via Lusternik-Schnirelman theory, and extra ones in case of double bubbles whose opposite boundaries have the same mean curvature.