I have already blogged several times about complete Riemannian metrics of positive scalar curvature on 3-manifolds: here, here and here.

Now it seems that finally Jian Wang proved the result that he was working on for quite some time now (arXiv:2105.07095): *Any contractible 3-manifold admitting a complete Riemannian metric of non-negative scalar curvature is homeomorphic to Euclidean 3-space.*

edit (August 3rd, 2022): Unfortunately, Jian Wang retracted his result from the arXiv.

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