Deforming three-manifolds with positive scalar curvature

TL;DR

This paper proves that the space of positive scalar curvature metrics on orientable compact 3-manifolds is path-connected, using Ricci flow, conformal methods, and Gromov-Lawson constructions, with applications to Einstein constraint equations.

Contribution

It establishes the path-connectedness of the moduli space of positive scalar curvature metrics on 3-manifolds, combining Ricci flow techniques with classical geometric constructions.

Findings

The moduli space of positive scalar curvature metrics is path-connected.

The methods extend to solutions of Einstein constraint equations.

Utilizes Perelman's work on Ricci flow as a fundamental tool.

Abstract

In this paper we prove that the moduli space of metrics with positive scalar curvature of an orientable compact 3-manifold is path-connected. The proof uses the Ricci flow with surgery, the conformal method, and the connected sum construction of Gromov and Lawson. The work of Perelman on Hamilton's Ricci flow is fundamental. As one of the applications we prove the path-connectedness of the space of trace-free asymptotically flat solutions to the vacuum Einstein constraint equations on $\mathbb{R}^3$.