Deforming 3-manifolds of bounded geometry and uniformly positive scalar curvature

TL;DR

This paper proves that the space of complete 3-manifold metrics with bounded geometry and positive scalar curvature is path-connected, extending previous compact case results using Ricci flow and geometric control techniques.

Contribution

It generalizes the connectedness result of the moduli space to non-compact 3-manifolds with bounded geometry and positive scalar curvature.

Findings

The moduli space of such metrics is path-connected.

Extension of previous compact case results to non-compact manifolds.

Application of Ricci flow with surgery in the non-compact setting.

Abstract

We prove that the moduli space of complete Riemannian metrics of bounded geometry and uniformly positive scalar curvature on an orientable 3-manifold is path-connected. This generalizes the main result of the fourth author [Mar12] in the compact case. The proof uses Ricci flow with surgery as well as arguments involving performing infinite connected sums with control on the geometry.