On open manifolds admitting no complete metric with positive scalar curvature

Abstract: In this paper, we investigate the topological obstruction problem for positive scalar curvature and uniformly positive scalar curvature on open manifolds. We present a definition for open Schoen-Yau-Schick manifolds and prove that there is no complete metric with positive scalar curvature on these manifolds. Similarly, we define weak Schoen-Yau-Shick manifolds by analogy, which are expected to admit no complete metrics with uniformly positive scalar curvature.