Double bubbles in $S^3$ and $H^3$

TL;DR

This paper proves the double bubble conjecture in three-sphere and hyperbolic three-space under certain volume conditions, using Hutchings theory, asymptotic analysis, and computer verification.

Contribution

It extends the double bubble conjecture proof to $S^3$ and $H^3$ for specific volume ranges using a combination of theoretical and computational methods.

Findings

Double bubble conjecture proven in $S^3$ and $H^3$ under specified conditions.

Computer analysis confirms the conjecture in the given cases.

Reduction of the problem to computational checks via asymptotic and balancing arguments.

Abstract

We prove the double bubble conjecture in the three-sphere $S^3$ and hyperbolic three-space $H^3$ in the cases where we can apply Hutchings theory: 1) in $S^3$, each enclosed volume and the complement occupy at least 10% of the volume of $S^3$; 2) in $H^3$, the smaller volume is at least 85% that of the larger. A balancing argument and asymptotic analysis reduce the problem in $S^3$ and $H^3$ to some computer checking. The computer analysis has been designed and fully implemented for both spaces.