3-Manifolds with positive scalar curvature and bounded geometry

TL;DR

This paper proves that complete contractible 3-manifolds with positive scalar curvature and bounded geometry are topologically equivalent to ^3, and that higher genus handlebodies cannot admit such metrics, using inverse mean curvature flow techniques.

Contribution

It establishes topological restrictions on 3-manifolds with positive scalar curvature and bounded geometry, extending understanding of geometric conditions.

Findings

Complete contractible 3-manifolds with positive scalar curvature are ^3.

Handlebodies of genus > 1 cannot have complete metrics with positive scalar curvature and bounded geometry.

Results are based on inverse mean curvature flow methods.

Abstract

We show that a complete contractible 3-manifold with positive scalar curvature and bounded geometry must be $\mathbb R^3$. We also show that an open handlebody of genus larger than 1 does not admit complete metrics with positive scalar curvature and bounded geometry. Our results rely on the maximal weak solution to inverse mean curvature flow due to the third-named author.