arXiv subjects

Tin-Yau Tsang

Publications and source records attributed to Tin-Yau Tsang.

7 recordsLinked to original sources

arXiv2026

Positive mass theorems for manifolds with asymptotically locally hyperbolic toroidal ends

In [Classical Quantum Gravity 35, 115015 (2018)], P. Chruściel, L. Nguyen, T.-T. Paetz and the first-named author obtained a positive mass theorem 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 positive mass 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 $μ$-bubbles) introduced in work of D. A. Lee, M. Lesourd, and R. Unger [Calculus Var. Part. Differ. Equations 62(7), 194 (2023)] for manifolds with an asymptotically flat end, and further developed by the second-named author in [arXiv:2604.26978 (2026)] for manifolds with an asymptotically hyperbolic end.

math.DG↗

arXiv2026

Positive mass theorem for initial data sets with arbitrary ends

We showed a positive energy theorem for asymptotically flat initial data sets with the concept of spectral PSC by He-Shi-Yu, Bi-Hao-He-Shi-Zhu and Brendle-Wang; and the Jang equation in Schoen-Yau, Eichmair and Jang. Then, we proved a quantitative shielding theorem concerning the causal property of the energy-momentum vector of an asymptotically hyperbolic manifold. As a result, we established the positive mass theorem for complete asymptotically hyperbolic manifolds satisfying the dominant energy condition. As corollaries, we also obtained corresponding results for manifolds with asymptotically locally hyperbolic ends with a certain symmetry.

math.DG↗

arXiv2024

Remarks on quasilocal mass and fill-ins

In this paper we would have a brief overview of several proposals of quasilocal mass which are based on Hamiltonian formulation. We also show the positivity of the Wang-Yau energy under a more general condition. We then further study the quasilocal mass and DEC fill-ins defined by the author in terms of completeness and shields.

math.DG↗

arXiv2022

Monotonicity of the $p$-Green functions

On a complete $p$-nonparabolic $3$-dimensional manifold with non-negative scalar curvature and vanishing second homology, we establish a sharp monotonicity formula for the proper $p$-Green function along its level sets for $1<p<3$. This can be viewed as a generalization of the recent result by Munteanu-Wang \cite{MunteanuWang2021} in the case of $p=2$. No smoothness assumption is made on the $p$-Green function when $1<p\leq 2$. Several rigidity results are also proven.

math.AP↗

arXiv2022

On a spacetime positive mass theorem with corners

In this paper we consider the positive mass theorem for general initial data sets satisfying the dominant energy condition which are singular across a piecewise smooth surface. We find jump conditions on the metric and second fundamental form which are sufficient for the positivity of the total spacetime mass. Our method extends that of Hirsch-Kazaras-Khuri to the singular case (which we refer to as initial data sets with corners) using some ideas from Hirsch-Miao-Tsang. As such we give an integral lower bound on the spacetime mass and we characterise the case of zero mass. Our approach also leads to a new notion of quasilocal mass which we show to be positive, extending the work of Shi-Tam to the spacetime case. Moreover, we give sufficient conditions under which spacetime Bartnik data sets cannot admit a fill-in satisfying the dominant energy condition. This generalises the work of Shi-Wang-Wei-Zhu and Shi-Wang-Wei to the spacetime setting.

math.DG↗

arXiv2021

Dihedral rigidity for cubic initial data sets

In this paper we pose and prove a spacetime version of Gromov's dihedral rigidity theorem (Gromov, Li) for cubes when the dimension is 3 by studying the level sets of spacetime harmonic functions (Stern, Bray-Stern, Hirsch-Kazaras-Khuri), extending the work of Chai-Kim. As a corollary, we also obtain an alternative proof of dihedral rigidity for prisms in hyperbolic space (Li). We then discuss the relation between polyhedra and the spacetime positive mass theorem. This generalises the work of Miao-Piubello and Li. Finally, we show dihedral rigidity of charged Riemannian cubes by charged harmonic functions (Bray-Hirsch-Kazaras-Khuri-Zhang).

math.DG↗

arXiv2021

Mass of asymptotically flat $3$-manifolds with boundary

We study the mass of asymptotically flat $3$-manifolds with boundary using the method of Bray-Kazaras-Khuri-Stern. More precisely, we derive a mass formula on the union of an asymptotically flat manifold and fill-ins of its boundary, and give new sufficient conditions guaranteeing the positivity of the mass. Motivation to such consideration comes from studying the quasi-local mass of the boundary surface. If the boundary isometrically embeds in the Euclidean space, we apply the formula to obtain convergence of the Brown-York mass along large surfaces tending to $\infty$ which include the scaling of any fixed coordinate-convex surface.

math.DG↗