A complete metric of uniformly positive scalar curvature forces a suitably connected contractible open 5-manifold to be R^5, and positive isotropic curvature or pinched Ricci conditions with convex boundary force compact contractible manifolds to be disks.
3-Manifolds with positive scalar curvature and bounded geometry
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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.
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Positive curvature conditions on contractible manifolds
A complete metric of uniformly positive scalar curvature forces a suitably connected contractible open 5-manifold to be R^5, and positive isotropic curvature or pinched Ricci conditions with convex boundary force compact contractible manifolds to be disks.