Positive mass theorems for manifolds with ALH toroidal ends

TL;DR

This paper extends positive mass theorems to manifolds with ALH toroidal ends without boundary, using advanced techniques involving MOTS and $mbda$-bubbles, broadening the class of manifolds where positive mass results hold.

Contribution

It introduces new positive mass theorems for ALH manifolds with more general ends, employing sophisticated MOTS-based and $mbda$-bubble techniques.

Findings

Positive mass theorems established for manifolds with ALH toroidal ends without boundary.

Generalization of previous results to broader classes of asymptotic geometries.

Use of $mbda$-bubbles technique in the context of positive mass theorems.

Abstract

In work with P. Chru\'sciel, L. Nguyen and T.-T. Paetz [8], a positive mass theorem was obtained for asymptotically locally hyperbolic manifolds with boundary, having a toroidal end. The proof made use of properties of marginally outer trapped surfaces (MOTS). Here we present some new PMT results for such manifolds, but without boundary, which allow for other more general ends. The proofs, while still MOTS-based, involve a more elaborate technique (related to $\mu$-bubbles) introduced in work of D. A. Lee, M. Lesourd, and R. Unger [20] for manifolds with an asymptotically flat end, and further developed in [23] for manifolds with an asymptotically hyperbolic end.