Complete noncompact G2-manifolds with ALC asymptotics

TL;DR

This paper establishes existence, uniqueness, and structure results for complete noncompact G2-holonomy manifolds with ALC asymptotics, drawing parallels with hyperk"ahler geometry and developing a robust Fredholm theory.

Contribution

It introduces a Fredholm theory for ALC spaces applicable to G2-holonomy metrics, including a G2-analogue of the Atiyah-Hitchin metric and a moduli space framework.

Findings

Existence of a G2-analogue of the Atiyah-Hitchin metric

Development of a Fredholm theory for ALC spaces

Rigidity results based on asymptotic symmetries

Abstract

We prove existence, uniqueness and structure results for complete noncompact 7-dimensional G2-holonomy metrics with ALC (asymptotically locally conical) asymptotics. We regard such spaces as G2-analogues of ALF gravitational instantons in 4-dimensional hyperk\"ahler geometry. Our main results include the existence of a G2-analogue of the Atiyah-Hitchin metric in 4-dimensional hyperk\"ahler geometry, the existence of a good moduli theory for ALC G2-holonomy metrics and rigidity results for ALC G2-metrics in terms of the symmetries of their asymptotic model. The analytic toolkit needed to prove all these results is a robust Fredholm theory for the natural geometric linear elliptic operators on ALC spaces. We provide a self-contained derivation of this Fredholm theory for arbitrary Riemannian manifolds with ALC asymptotics. Since our ALC Fredholm theory does not rely on imposing any holonomy reduction or curvature conditions it may also be of utility beyond the setting of ALC special holonomy metrics. As one such application of our general Fredholm theory we prove some Hodge-theoretic results on general ALC spaces.