The asymptotic geometry of $\rm G_2$-monopoles

Abstract

This article investigates the asymptotics of $\rm G_2$-monopoles. First, we prove that when the underlying $\rm G_2$-manifold has polynomial volume growth strictly faster than $r^{7/2}$, 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.Comment: 80 pages. v2: added 4 new sections, new results, including a third main result; previous sections fully revised, exposition improved, corrected typos and reworked some proof