A Positive Mass Type Theorem for a Singular Toroidal Static Model
Abstract
We study positive mass and related Brown--York type inequalities for three-dimensional manifolds modeled on a static space with flat toroidal slices. Using inverse mean curvature flow, we first derive an inequality relating the asymptotic geometry at infinity to the size of an interior singularity. This inequality admits a natural interpretation as a positive mass type theorem. We then combine this global inequality with a Shi--Tam type construction to obtain a Brown--York type inequality for boundaries isometric to flat tori. In contrast to the Schwarzschild setting, our argument does not extend directly to general boundary surfaces, and we identify several obstructions to such a generalization. Finally, we discuss examples illustrating the relationship between negative mass, the strength of interior singularities, and the topology of the interior.
Related
- Positive mass theorems for singular asymptotically hyperbolic manifolds
- Gromov-Hausdorff instability of the positive mass theorem for Kähler manifolds
- A continuous Positive Mass Theorem for perturbations of Euclidean space
- Stability of the Riemannian positive mass theorem in all dimensions
- Rigidity of positive mass theorem with fast metric decay
BibTeX
@misc{wang2026positive,
title = {{A Positive Mass Type Theorem for a Singular Toroidal Static Model}},
author = {Zhixin Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27444},
eprint = {2609.27444},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27444},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - A Positive Mass Type Theorem for a Singular Toroidal Static Model
AU - Zhixin Wang
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.27444
UR - https://arxiv.org/abs/2609.27444
AB - We study positive mass and related Brown--York type inequalities for three-dimensional manifolds modeled on a static space with flat toroidal slices. Using inverse mean curvature flow, we first derive an inequality relating the asymptotic geometry at infinity to the size of an interior singularity. This inequality admits a natural interpretation as a positive mass type theorem. We then combine this global inequality with a Shi--Tam type construction to obtain a Brown--York type inequality for boundaries isometric to flat tori. In contrast to the Schwarzschild setting, our argument does not extend directly to general boundary surfaces, and we identify several obstructions to such a generalization. Finally, we discuss examples illustrating the relationship between negative mass, the strength of interior singularities, and the topology of the interior.
ER -