Ricci-flat deformations and special holonomy

TL;DR

This paper proves that the space of Ricci-flat metrics arising from torsion-free G-structures on compact manifolds with special holonomy groups is smooth and open, extending results to asymptotically cylindrical cases for Spin(7) and G_2.

Contribution

It establishes the openness and smoothness of the moduli space of Ricci-flat metrics from torsion-free G-structures, including extensions to asymptotically cylindrical manifolds for certain groups.

Findings

The natural map from G-structure moduli to Ricci-flat metrics is open.

The image of this map forms a smooth manifold.

Results extend to asymptotically cylindrical manifolds for Spin(7) and G_2 cases.

Abstract

Let G be one of the Ricci-flat holonomy groups SU(n), Sp(n), Spin(7) or G_2, and M a compact manifold of dimension 2n, 4n, 8 or 7, respectively. We prove that the natural map from the moduli space of torsion-free G-structures on M to the moduli space of Ricci-flat metrics is open, and that the image is a smooth manifold. For the exceptional cases G = Spin(7) and G_2 we extend the result to asymptotically cylindrical manifolds.