Triple bubbles in the hyperbolic plane

Abstract

We prove that the standard triple bubble uniquely minimizes perimeter among clusters in the hyperbolic plane with three prescribed positive finite areas. The standard family admits global pressure coordinates, and its volume differential is the negative of a positive definite grounded weighted graph Laplacian. A normalized perimeter excess attains a comparison minimum under a perimeter bound and gives a strict comparison between actual and model pressures. Component relabeling reduces the possible topologies to an all-boundary network or a 4-daisy. Discrete comparisons exclude the 4-daisy; pressure-preserving face deletion reduces the remaining network to the standard tetrahedral configuration.