Deformations of asymptotically cylindrical G_2 manifolds

TL;DR

This paper investigates the deformation theory of asymptotically cylindrical G_2 manifolds, establishing smoothness of their moduli space and characterizing when their holonomy is exactly G_2 based on topological conditions.

Contribution

It proves the smoothness of the moduli space of torsion-free G_2 structures on asymptotically cylindrical manifolds and characterizes holonomy conditions in terms of fundamental group and topology.

Findings

Moduli space of G_2 structures is smooth if non-empty.

Holonomy is G_2 iff fundamental group is finite and no double cover is cylindrical.

Provides local properties of the moduli space.

Abstract

We prove that for a 7-dimensional manifold M with cylindrical ends the moduli space of exponentially asymptotically cylindrical torsion-free G_2 structures is a smooth manifold (if non-empty), and study some of its local properties. We also show that the holonomy of the induced metric of an exponentially asymptotically cylindrical G_2 manifold M is exactly G_2 if and only if its fundamental group is finite and neither M nor any double cover of M is homeomorphic to a cylinder.