Constrained deformations of positive scalar curvature metrics, II

TL;DR

This paper proves the contractibility of certain spaces of positive scalar curvature metrics on 3-manifolds with boundary under boundary mean curvature constraints, impacting the topology of initial data sets in general relativity.

Contribution

It establishes the contractibility of spaces of constrained positive scalar curvature metrics on 3-manifolds, including mean-convex and minimal boundary cases, with implications for Einstein's equations.

Findings

Spaces of constrained positive scalar curvature metrics are contractible.

Results apply to both mean-convex and minimal boundary conditions.

Implications for the topology of initial data sets in general relativity.

Abstract

We prove that various spaces of constrained positive scalar curvature metrics on compact 3-manifolds with boundary, when not empty, are contractible. The constraints we mostly focus on are given in terms of local conditions on the mean curvature of the boundary, and our treatment includes both the mean-convex and the minimal case. We then discuss the implications of these results on the topology of different subspaces of asymptotically flat initial data sets for the Einstein field equations in general relativity.