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.
Ricci flow with surgery on manifolds with positive isotropic curva- ture
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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.