Topology of 3-manifolds with nonnegative scalar curvature and positive harmonic functions

TL;DR

This paper investigates the topology of complete 3-manifolds with nonnegative scalar curvature, showing contractible cases are diffeomorphic to Euclidean space and bounding the genus of handlebodies.

Contribution

It establishes new topological classifications for 3-manifolds with nonnegative scalar curvature using harmonic functions and gradient estimates.

Findings

Contractible 3-manifolds with nonnegative scalar curvature are diffeomorphic to ^3.

Open handlebodies with such metrics have genus at most 1.

The proof employs level set exhaustions and refined gradient estimates.

Abstract

We study complete $3$-manifolds with nonnegative scalar curvature under additional regularity assumptions. We prove that a contractible such manifold is diffeomorphic to $\mathbb{R}^3$, and that an open handlebody admitting such a metric must have genus at most $1$. The proof uses exhaustions by level sets of harmonic functions and refined average gradient estimates.