The topological structure of contractible 3-manifolds is much complicated. For example, the Whitehead manifold is a contractible 3-manifold but not homeomorphic to $\mathbf{R}3$. In the talk, we will use minimal surfaces theory to present a proof that the Whitehead manifold has no complete metric with positive scalar curvature. Further, we show that a complete contractible 3-manifold with positive scalar curvature is homeomorphic to $\mathbf{R} 3$.
Links:
[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/node/3050