What question did this study set out to answer?

The aim is to explore the topological limits on the existence of complete metrics with positive scalar curvature in open manifolds.

On open manifolds admitting no complete metric with positive scalar curvature

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The exploration investigates metrics with positive scalar curvature in open manifolds, indicating significant topological constraints.

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.

Shi et al. (2026) studied this question.

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