In this article, we classify (non-compact) 3 3 -manifolds with uniformly positive scalar curvature. Precisely, we show that an orientable 3-manifold has a complete metric with uniformly positive scalar curvature if and only if it is homeomorphic to a connected sum of spherical 3 3 -manifolds and some copies of S 2 × S 1 S² S¹. Further, we study a 3 3 -manifold with mean convex boundary and with uniformly positive scalar curvature. If the boundary is a disjoint union of closed surfaces, then the manifold is a connected sum of spherical 3 3 -manifolds, some copies of S 1 × S 2 S¹ S² and some handlebodies.
Jian Wang (Wed,) studied this question.