The Hitchin and Sutcliffe metrics for hyperbolic 2-monopoles

Abstract

We compare two currently known examples of hyperbolic SU(2) two-monopole metrics and their candidacy as models of hyperbolic monopole dynamics. The first is Sutcliffe's boundary metric and the other is a well-known self-dual Einstein metric found by Hitchin. We show that these metrics are conformally inequivalent and make significantly different predictions for monopole dynamics. We compare each metric to the Franchetti-Ross point particle approximation of hyperbolic monopole dynamics. We show Sutcliffe's metric does not converge to the point particle approximation in the large separation limit. We also check Sen's conjecture for each metric by computing their $L^2$ harmonic two-forms. By a symmetry argument we are able to show that Hitchin's metric has a single $L^2$ harmonic two-form consistent with the conjecture.