On noncollapsed almost Ricci-flat 4-manifolds

TL;DR

This paper establishes topological criteria under which noncollapsed almost Ricci-flat 4-manifolds can be shown to admit Ricci-flat metrics, linking geometry with topology.

Contribution

It introduces new topological conditions, such as spin structure with nonzero A-hat genus or large fundamental group, guaranteeing Ricci-flat metrics on these manifolds.

Findings

Spin manifolds with nonzero A-hat genus admit Ricci-flat metrics.

Manifolds with infinite or large fundamental groups admit Ricci-flat metrics.

Topological conditions are sufficient for the existence of Ricci-flat metrics.

Abstract

We give topological conditions to ensure that a noncollapsed almost Ricci-flat 4-manifold admits a Ricci-flat metric. One sufficient condition is that the manifold is spin and has a nonzero A-hat genus. Another condition is that the fundamental group is infinite or, more generally, of sufficiently large cardinality.