Contractible $3$-manifold and Positive scalar curvature (I)

TL;DR

This paper proves that the Whitehead manifold and similar genus one contractible 3-manifolds cannot support complete metrics with positive scalar curvature, extending known results in geometric topology.

Contribution

It establishes the non-existence of positive scalar curvature metrics on Whitehead and genus one contractible 3-manifolds, generalizing previous results.

Findings

Whitehead manifold admits no complete positive scalar curvature metric

Genus one contractible 3-manifolds also cannot support such metrics

Extends understanding of scalar curvature constraints in 3-manifold topology

Abstract

In this work we prove that the Whitehead manifold has no complete metric of positive scalar curvature. This result can be generalized to the genus one case. Precisely, we show that no contractible genus one $3$-manifold admits a complete metric of positive scalar curvature.