Positive curvature conditions on contractible manifolds

TL;DR

This paper explores curvature conditions that uniquely identify Euclidean space and disks among contractible manifolds, showing that certain positive scalar curvature conditions imply the manifold is standard, with some conditions requiring strengthening.

Contribution

It establishes that positive scalar curvature on certain contractible manifolds implies they are diffeomorphic to Euclidean space or disks, and identifies stronger conditions needed for this conclusion.

Findings

Positive scalar curvature implies Euclidean space in certain open manifolds.

Positive scalar curvature alone does not characterize disks among compact manifolds.

Stronger curvature conditions can distinguish disks from other contractible manifolds.

Abstract

Our goal is to identify curvature conditions that distinguish Euclidean space in the case of open, contractible manifolds and the disk in the case of compact, contractible manifolds with boundary. First, we show that an open manifold that is the interior of a sufficiently connected, compact, contractible 5-manifold with boundary and supports a complete Riemannian metric with uniformly positive scalar curvature is diffeomorphic to Euclidean 5-space. Next, we investigate the analogous question for compact manifolds with boundary: Must a compact, contractible manifold that supports a Riemannian metric with positive scalar curvature and mean convex boundary necessarily be the disk? We present examples demonstrating that this curvature condition alone cannot distinguish the disk; on the other hand, we exhibit stronger curvature conditions that allow us to draw such a conclusion.