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

Abstract

This article investigates the asymptotics of ${\mathrm{G}}_2$-monopoles. First, we prove that when the underlying ${\mathrm{G}}_2$-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 ${\mathrm{G}}_2$-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 ${\mathrm{G}}_2$-manifold.

83 pages. v4: Corrected minor mistakes/typos on the Bochner--Weitzenböck formulas of Lemmas 5.2, 5.3, 5.5 and 9.1. In particular, this led to a restatement of both Corollary 5.4 and Proposition 6.3, and their proofs were rewritten. Final version to appear in the Memoirs of the American Mathematical Society

References in corpus (2)

• G2 instantons and the Seiberg-Witten monopoles
• Sigma Models with Repulsive Potentials

Cited by in corpus (1)

• $SU(2{)}^2$-invariant gauge theory on asymptotically conical Calabi-Yau 3-folds