On open manifolds admitting no complete metric with positive scalar curvature

Key Points

• The aim is to explore the topological limits on the existence of complete metrics with positive scalar curvature in open manifolds.
• Introduction of open Schoen–Yau–Schick manifolds.
• Definition of weak Schoen–Yau–Shick manifolds.
• Proof of the absence of complete metrics with positive scalar curvature.
• No complete metric with positive scalar curvature can exist on open Schoen–Yau–Schick manifolds.
• Weak Schoen–Yau–Shick manifolds are likely to also lack complete metrics with uniformly positive scalar curvature.

Abstract