Paper · Sep 23, 2026
A Positive Mass Type Theorem for a Singular Toroidal Static Model
As on arXiv
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.
Its neighbourhood
Connected papers
The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.
-
On the geometry and uniqueness of asymptotically locally hyperbolic static vacuum black holes
Topics Inverse mean curvature flow, Physics, 3-manifolds · Shared terms “static space”, “toroidal”, “Schwarzschild”
-
Positive mass and isoperimetry for continuous metrics with nonnegative scalar curvature
Topics Inverse mean curvature flow, Physics, Other curvature flows · Shared terms “positive mass”
-
Hawking Mass Monotonicity for Initial Data Sets
Topics Inverse mean curvature flow, Physics, Other curvature flows
-
Scalar curvature bounds for 3D continuous metrics through the Inverse Mean Curvature Flow
Topics Inverse mean curvature flow, Physics, 3-manifolds
-
Monotone Quantities for -Harmonic functions and the Sharp -Penrose inequality
Topics Inverse mean curvature flow, Physics, 3-manifolds · Shared terms “relating”, “Schwarzschild”
-
The -inverse mean curvature flow and the generalized Penrose conjecture
Topics Inverse mean curvature flow, Physics, Other curvature flows
-
Classification of ruled surfaces as homothetic self-similar solutions of the inverse mean curvature flow in the Lorentz-Minkowski 3-space
Topics Inverse mean curvature flow, 3-manifolds, Other curvature flows
-
The Riemannian Penrose Inequality with Matter Density
Topics Inverse mean curvature flow, Physics, Other curvature flows
Read around it
Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
- 10A.1A Zoo of Three-Manifolds
The 3-sphere, lens spaces, the Poincaré sphere, the Hopf fibration and the shape of space.
- 10A.2Finite Rotation Groups and Spherical Space Forms
Klein's classification by counting poles, the binary groups, and every quotient of the round 3-sphere.
- 10A.4Homotopy Groups, Degree and the Hurewicz Theorem
Spheres mapped into manifolds: the algebraic topology that finite extinction runs on.