The asymptotic geometry of $\rm{G}_2$-monopoles

TL;DR

This paper studies the behavior and classification of $ m{G}_2$-monopoles on special 7-manifolds, establishing decay properties, mass finiteness, and a framework for their moduli space, especially on asymptotically conical geometries.

Contribution

It proves finiteness of mass for monopoles on nonparabolic $ m{G}_2$-manifolds, derives decay estimates and convergence results on asymptotically conical manifolds, and develops a Fredholm theory for their moduli space.

Findings

Finite mass for monopoles on nonparabolic $ m{G}_2$-manifolds.

Sharp decay estimates and convergence to pseudo-Hermitian--Yang--Mills connections.

Fredholm setup for the moduli space of monopoles.

Abstract

This article investigates the asymptotics of $\rm{G}_2$-monopoles. First, we prove that when the underlying $\rm{G}_2$-manifold is nonparabolic (i.e. admits a positive Green's function), finite intermediate energy monopoles with bounded curvature have finite mass. The second main result restricts to the case when the underlying $\rm{G}_2$-manifold is asymptotically conical. In this situation, we deduce sharp decay estimates and that the connection converges, along the end, to a pseudo-Hermitian--Yang--Mills connection over the asymptotic cone. Finally, our last result exhibits a Fredholm setup describing the moduli space of finite intermediate energy monopoles on an asymptotically conical $\rm{G}_2$-manifold.