Topological rigidity and Positive scalar curvature

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 $\mathbf{R}^3$.