Positive Scalar Curvature on Noncompact Manifolds and the Liouville Theorem

TL;DR

This paper establishes topological obstructions to positive scalar curvature on noncompact manifolds using minimal hypersurfaces, proves a Liouville theorem for conformally flat manifolds, and extends results via MOTS and energy conditions.

Contribution

It introduces new topological obstructions for noncompact manifolds and extends Liouville theorems using minimal hypersurfaces and MOTS techniques.

Findings

Topological obstructions to positive scalar curvature on certain noncompact manifolds.

Liouville theorem for conformally flat manifolds with non-negative scalar curvature.

Extension of results using MOTS and energy conditions.

Abstract

Using minimal hypersurfaces, we obtain topological obstructions to admitting complete metrics with positive scalar curvature on a given class of non-compact n-manifolds with n less than 8. We show that the Liouville theorem for a locally conformally flat n-manifold of non-negative scalar curvature follows from the impossibility of there being a positive scalar curvature metric on its connect sum with the n-torus. With the recent work of Chodosh-Li, the Liouville theorem is now proved in all remaining cases. Finally, using MOTS instead of minimal hypersurfaces, we show an Initial Data Set version of these results with the Dominant Energy Scalar appearing instead of positive scalar curvature.