An orientable 3-manifold with positive scalar curvature decaying at rate C > 2/3 must be a connected sum of spherical manifolds and S^2 x S^1 pieces, and the threshold 2/3 is optimal.
Positive curvature conditions on contractible manifolds
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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.
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Optimal decay constant for complete manifolds of positive scalar curvature with quadratic decay
An orientable 3-manifold with positive scalar curvature decaying at rate C > 2/3 must be a connected sum of spherical manifolds and S^2 x S^1 pieces, and the threshold 2/3 is optimal.