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.

March 28, 2026Open Access

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

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.

On open manifolds admitting no complete metric with positive scalar curvature

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Abstract

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