Double bubbles for immiscible fluids in $\mathbb{R}^n$

TL;DR

This paper proves that weighted double bubbles in n-dimensional Euclidean space minimize surface energy for immiscible fluids, introducing a unification approach and a new symmetry argument for minimizer characterization.

Contribution

It presents a novel unification method to prove minimality of weighted double bubbles and introduces a new symmetry argument for identifying minimizers.

Findings

Weighted double bubbles minimize surface energy in n-dimensional space.

The proof simplifies the understanding of the double bubble theorem.

A new symmetry argument shows minimizers are surfaces of revolution.

Abstract

We use a new approach that we call unification to prove that standard weighted double bubbles in $n$-dimensional Euclidean space minimize immiscible fluid surface energy, that is, surface area weighted by constants. The result is new for weighted area, and also gives the simplest known proof to date of the (unit weight) double bubble theorem. As part of the proof we introduce a striking new symmetry argument for showing that a minimizer must be a surface of revolution.