Density and positive mass theorems for incomplete manifolds

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].