# Desingularizing positive scalar curvature 4-manifolds

## TL;DR

This paper proves the triviality of the bordism group of 3-manifolds with positive scalar curvature using explicit constructions, and applies these results to study singular 4-manifolds, leading to a positive mass theorem in low regularity settings.

## Contribution

It introduces explicit methods for bordism triviality, studies singular psc 4-manifolds, and establishes a positive mass theorem for low-regularity asymptotically flat 4-manifolds.

## Key findings

- Trivial bordism group of closed 3-manifolds with psc.
- Construction of psc null-bordisms for singular 4-manifolds.
- Positive mass theorem for low-regularity asymptotically flat 4-manifolds.

## Abstract

We show that the bordism group of closed 3-manifolds with positive scalar curvature (psc) metrics is trivial by explicit methods. Our constructions are derived from scalar-flat K{\"a}hler ALE surfaces discovered by Lock-Viaclovsky. Next, we study psc 4-manifolds with metric singularities along points and embedded circles. Our psc null-bordisms are essential tools in a desingularization process developed by Li-Mantoulidis. This allows us to prove a non-existence result for singular psc metrics on enlargeable 4-manifolds with uniformly Euclidean geometry. As a consequence, we obtain a positive mass theorem for asymptotically flat 4-manifolds with non-negative scalar curvature and low regularity.