It is known that \(\mathbb{R}^3\) admits a complete metric of uniformly positive scalar curvature.

In fact, for any closed manifold \(X\) and any \(k \ge 3\) the manifold \(X \times \mathbb{R}^k\) admits a complete metric of uniformly positive scalar curvature by a result of Rosenberg and Stolz (link).

Now there exist contractible, open 3-manifolds which are not homeomorphic to \(\mathbb{R}^3\), e.g., the Whitehead manifold. It was proven by Chang-Weinberger-Yu (link) that no such manifold can admit a complete metric of uniformly positive scalar curvature.

In a recent preprint (arXiv:1805.03544) Jian Wang improved the above stated result of Chang-Weinberger-Yu. He proved that if a contractible 3-manifold admits a complete metric whose scalar curvature is positive and decays at most like \(d(x_0, -)^\alpha\), where \(\alpha \in [0,2)\) and \(x_0\) is a point in the manifold, then the manifold must be homeomorphic to \(\mathbb{R}^3\).

The basic ingredient in Wang’s proof is the theorem of Stallings that a contractible 3-manifold is homeomorphic to \(\mathbb{R}^3\) if and only if it is simply-connected at infinity. So assuming that the contractible 3-manifold is not simply-connected at infinity, Wang makes a clever choice of curve witnessing this and applies the solution of the Plateau problem to it, i.e., fills the curve with an area minizing disc. The scalar curvature condition comes in due to a result of Rosenberg (link) that this disc must be close to its boundary, i.e., to the chosen curve, contradicting the clever choice of that curve (since the filling constitutes a null-homotopy of the curve which is close to the curve itself, but the curve is chosen such that one has to go far away from it to contract it).

*edit (23.08.2020):* It seems that Stallings is the wrong reference for the stated result about 3-manifolds. Stallings proved it in the case of contractible manifolds of dimension 5 and higher. In dimension 4 it is due to Guilbault, and in dimension 3 due to Husch and Price together with Perelman. The concrete references may be found in this MathOverflow answer: link.

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