Let \((M,g)\) be a complete, contractible Riemannian \(3\)-manifold (without boundary).
- Chang-Weinberger-Yu (link) proved that if \((M,g)\) has uniformly positive scalar curvature, then \(M\) must be homeomorphic to \(\mathbb{R}^3\).
- Recently (arXiv:1906.04128), Wang proved that if \((M,g)\) has positive scalar curvature and \(M\) has trivial fundamental group at infinity, then \(M\) must be homeomorphic to \(\mathbb{R}^3\).
Jiang Wang already has some results on complete metrics with positive scalar curvature on contractible \(3\)-manifolds:
- I blogged here about his generalization of the result of Chang-Weinberger-Yu (a sufficiently slow decay at infinity of the positive scalar curvature suffices to conclude that the manifold must be homeomorphic to \(\mathbb{R}^3\)).
- And I blogged here about his result that the Whitehead manifold does not admit any complete metric of positive scalar curvature. Note that the Whitehead manifold is a contractible \(3\)-manifold which is not homeomorphic to \(\mathbb{R}^3\). Since the Whitehead manifold has trivial fundamental group at infinity, his recent result generalizes this one.
Note that having trivial fundamental group at infinity is not the same as being simply-connected at infinity. Stallings proved that a contractible 3-manifold is homeomorphic to \(\mathbb{R}^3\) if and only if it is simply-connected at infinity. So corresponding version of Wang’s recent result with the condition of having trivial fundamental group at infinity replaced by being simply-connected at infinity is true without even having to consider the scalar curvature of the metric.
For the proof of his recent result Wang uses again minimal surfaces. It seems that this is a powerful technique to investigate complete metrics of positive scalar curvature on non-compact (\(3\)-dimensional) manifolds.