Density and positive mass theorems for incomplete manifolds

TL;DR

This paper establishes density theorems for incomplete manifolds with asymptotically flat ends, improving positive mass theorem results by allowing for incompleteness and negative scalar curvature away from the end.

Contribution

It introduces a density theorem for manifolds with asymptotically flat ends that accommodates incompleteness and negative scalar curvature, enhancing positive mass theorem applications.

Findings

Proves a density theorem producing harmonic asymptotics on the end.

Improves the quantitative positive mass theorem in dimensions 3 to 7.

Provides an alternative proof using MOTS and μ-bubbles.

Abstract

For manifolds with a distinguished asymptotically flat end, we prove a density theorem which produces harmonic asymptotics on the distinguished end, while allowing for points of incompleteness (or negative scalar curvature) away from this end. We use this to improve the "quantitative" version of the positive mass theorem (in dimensions $3\leq n\leq 7$), obtained by the last two named authors with S.-T. Yau [LUY21], where stronger decay was assumed on the distinguished end. We also give an alternative proof of this theorem based on a relationship between MOTS and $\mu$-bubbles and our recent work on the spacetime positive mass theorem with boundary [LLU21].