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August 17, 2025

Topology of complete 3-manifolds with uniformly positive scalar curvature

Key Points

An orientable 3-manifold has uniformly positive scalar curvature if it is homeomorphic to specific geometric sums, and some handlebodies.
Homeomorphic 3-manifolds can be expressed as a connected sum of spherical 3-manifolds and copies of products of spheres.
Observational analysis focused on manifolds with mean convex boundaries, leading to classification insights for 3-manifolds.
Findings indicate significant geometric implications for surfaces, warranting further exploration of 3-manifold structures.

Abstract

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.

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