Topological Rigidity of Contractible 3-Manifolds and Handlebody Interiors under Nonnegative Scalar Curvature

Abstract

We prove that a contractible $3$-manifold admitting a complete Riemannian metric of nonnegative scalar curvature is diffeomorphic to ${\mathbb{R}}^3$. We also prove that, if the interior $M_{\gamma }$ of a compact handlebody of genus $\gamma$ admits such a metric, then $\gamma \le 1$. This answers two open questions posed by Wang and Gromov, respectively.

36 Pages