Complete Riemannian 4-manifolds with uniformly positive scalar curvature

2024/07/08 by Chodosh, Otis, Maximo, Davi, Mukherjee, Anubhav · 3 citations
#Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT) #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.2407.05574

Abstract

We obtain topological obstructions to the existence of a complete Riemannian metric with uniformly positive scalar curvature on certain (non-compact) 4-manifolds. In particular, such a metric on the interior of a compact contractible 4-manifold uniquely distinguishes the standard 4-ball up to diffeomorphism among Mazur manifolds and up to homeomorphism in general. We additionally show there exist uncountably many exotic ℝ⁴'s that do not admit such a metric and that any (non-compact) tame 4-manifold has a smooth structure that does not admit such a metric.

Cited by

• Minimal surfaces and comparison geometry
• Optimal decay constant for complete manifolds of positive scalar curvature with quadratic decay
• Positive curvature conditions on contractible manifolds

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