A positive mass theorem for asymptotically flat manifolds with a non-compact boundary

TL;DR

This paper establishes a positive mass theorem for certain asymptotically flat manifolds with non-compact boundaries, extending previous results to higher dimensions and spin manifolds, with implications for geometric flows.

Contribution

It proves a positive mass theorem for asymptotically flat manifolds with non-compact boundary in specific dimensions and spin cases, addressing a question related to Yamabe-type flows.

Findings

Positive mass theorem proven for dimensions 3 to 7.

Extension to higher dimensions for spin manifolds.

Addresses a question on the behavior of Yamabe-type flows.

Abstract

We prove a positive mass theorem for $n$-dimensional asymptotically flat manifolds with a non-compact boundary if either $3\leq n\leq 7$ or if $n\geq 3$ and the manifold is spin. This settles, for this class of manifolds, a question posed in a recent paper by the first author in connection with the long-term behavior of a certain Yamabe-type flow on scalar-flat compact manifolds with boundary.