arXiv research

Compact method proves Brown-York mass positivity and connects to major conjectures.

Study of Brown--York mass for four-dimensional asymptotically flat manifolds.

In this paper, we will show that the limit of the Brown-York mass of a family of convex revolution surfaces in an asymptotically Schwarzschild manifold is the ADM mass.

In this paper, we will study the limiting behavior of the Brown-York mass of the coordinate spheres in an asymptotically flat manifold. Limiting behaviors of volumes of regions related to coordinate spheres are also obtained, including a discussion on the isoperimetric mass introduced by Huisken \cite{Huisken}. We will…

In this short note, we prove positivity of Brown-York mass under quasi-positive boundary data which generalize some previous results by the authors. The corresponding rigidity result is obtained.

We will extend partially our previous results about the limit of the Brown-York mass of a family of convex revolution surfaces in the Schwarzschild manifold such that these surfaces may have unbounded ratios of their radii.

We give some lower estimates of the ADM mass of an asymptotically flat (AF) Riemannian manifold without assuming that the scalar curvature of the manifold is nonnegative. Some sufficient conditions for an AF manifold to have nonnegative ADM mass are obtained. We also give some lower estimates of the Brown-York mass of …

We discuss some geometric problems related to the definitions of quasilocal mass proposed by Brown-York \cite{BYmass1} \cite{BYmass2} and Liu-Yau \cite{LY1} \cite{LY2}. Our discussion consists of three parts. In the first part, we propose a new variational problem on compact manifolds with boundary, which is motivated …

Compact proof for Brown-York mass positivity and rigidity in flat and spherical spaces.

The paper proves stability of a quasi-local positive mass theorem for graphical hypersurfaces.

We show that the limit at infinity of the vector-valued Brown-York-type quasi-local mass along any coordinate exhaustion of an asymptotically hyperbolic $3$-manifold satisfying the relevant energy condition on the scalar curvature has the conjectured causal character. Our proof uses spinors and relies on a Witten-type …

Study the mass of flat 3-manifolds with boundary using specific methods.

The Besse's conjecture was posted on the well-known book Einstein manifolds by Arthur L. Besse, which describes the critical point of Hilbert-Einstein functional with constraint of unit volume and constant scalar curvature. In this article, we show that there is an interesting connection between Besse's conjecture and …

On asymptotically flat and asymptotically hyperbolic manifolds, by evaluating the total mass via the Ricci tensor, we show that the limits of certain Brown-York type and Hawking type quasi-local mass integrals equal the total mass of the manifold in all dimensions.

Study geometric inequalities and boundary estimates for Einstein-type manifolds with boundary.

In this article, we investigate the connection between scalar curvature and first eigenfunctions via positive mass theorem for Brown-York mass. For compact manifolds with nice boundary, we show that a sharp inequality holds for first eigenfunctions when posing appropriate assumptions on scalar curvature and first eigen…

We study the supremum of the total mean curvature on the boundary of compact, mean-convex 3-manifolds with nonnegative scalar curvature, and a prescribed boundary metric. We establish an additivity property for this supremum and exhibit rigidity for maximizers assuming the supremum is attained. When the boundary consis…

Generalizes nonnegativity result for Brown-York mass using noncompact fill-ins.

In this paper, we will show that the limit of some quasilocal mass integrals of the coordinate spheres in an asymptotically hyperbolic (AH) manifold is the mass integral of the AH manifold. This is the analogue of the well known result that the limit of the Brown-York mass of coordinate spheres is the ADM mass in an as…

Study on quasi-Einstein manifolds with boundary estimates and inequalities.

In this paper, we obtain lower bounds for the Brown-York quasilocal mass and the Bartnik quasilocal mass for compact three manifolds with smooth boundaries. As a consequence, we derive sufficient conditions for the existence of horizons for a certain class of compact manifolds with boundary and some asymptotically flat…

In this article, we found a connection between Brown-York mass and the first Dirichlet Eigenvalue of a Schrödingier type operator. In particular, we proved a local positive mass type theorem for metrics conformal to the background one with suitable presumptions. As applications, we investigated compactly conformal defo…

In this paper, we study the limiting behavior of the Brown-York mass and Hawking mass along nearly round surfaces at infinity of an asymptotically flat manifold. Nearly round surfaces can be defined in an intrinsic way. Our results show that the ADM mass of an asymptotically flat 3-manifold can be approximated by some …

Positive mass theorem for tori with scalar curvature bounds.

Paper proves stability of positive mass theorem for specific types of manifolds.

Motivated by the quasi-local mass problem in general relativity, we apply the asymptotically flat extensions, constructed by Shi and Tam in the proof of the positivity of the Brown--York mass, to study a fill-in problem of realizing geometric data on a 2-sphere as the boundary of a compact 3-manifold of nonnegative sca…

We derive new inequalities between the boundary capacity of an asymptotically flat 3-manifold with nonnegative scalar curvature and boundary quantities that relate to quasi-local mass; one relates to Brown--York mass and the other is new. We argue by recasting the setup to the study of mean-convex fill-ins with nonnega…

In this paper lower bounds are obtained for quasi-local masses in terms of charge, angular momentum, and horizon area. In particular we treat three quasi-local masses based on a Hamiltonian approach, namely the Brown-York, Liu-Yau, and Wang-Yau masses. The geometric inequalities are motivated by analogous results for t…

Reviewed and extended Wang-Yau quasi-local mass definitions.

New mass inequalities and proofs for causal variational principles.

The paper proves nonexistence of NNSC cobordism for Bartnik data under certain conditions.

Defines a new quasi-local mass related to spacetime harmonic functions.

In this paper, we show that the Chen-Nester-Tung (CNT) quasi-local energy is closely related to the Wang-Yau (WY) quasi-local mass. As a particular example, we compute the second variation of the CNT quasi-local energy for axially symmetric Kerr-like spacetimes with axially symmetric embeddings at the obvious critical …

We use an idea of Wang and Yau to give a new definition of quasi-local mass for a topological sphere in an initial date set. The new definition modifies Brown-York's definition by using certain spinor norm as lapse function. And it requires mean curvature of the topological sphere satisfies apparent horizon conditions,…

Study on electrostatic systems with boundary, proving new geometric inequalities.

The article proves charged quasi-local Penrose inequalities for compact manifolds with boundary.

The paper proves conditions for positive scalar curvature metrics on manifolds with incompressible hypersurfaces.

We study a functional on the boundary of a compact Riemannian 3-manifold of nonnegative scalar curvature. The functional arises as the second variation of the Wang-Yau quasi-local energy in general relativity. We prove that the functional is positive definite on large coordinate spheres, and more general on nearly roun…

The definition of quasi-local mass for a bounded space-like region in space-time is essential in several major unsettled problems in general relativity. The quasi-local mass is expected to be a type of flux integral on the boundary two-surface and should be independent of whichever space-like region it bounds. An impor…

In this article, we study a free boundary isometric embedding problem for abstract Riemannian two-manifolds with the topology of the disc. Under the assumption of positive Gauss curvature and geodesic curvature of the boundary being equal to one, we show that any such disc may be isometrically embedded into the Euclide…

Study on static perfect fluid space-time geometry and boundary estimates.

Proves well-posedness for Einstein equations with specific boundary data.

Given a spacelike 2-surface $Σ$ in a spacetime $N$ and a constant future timelike unit vector $T_0$ in $\R^{3,1}$, we derive upper and lower estimates of Wang-Yau quasilocal energy $E(Σ, X, T_0)$ for a given isometric embedding $X$ of $Σ$ into a flat 3-slice in $\R^{3,1}$. The quantity $E(Σ, X, T_0)$ itself depends …

In this paper, we prove the following theorem regarding the Wang-Yau quasi-local energy of a spacelike two-surface in a spacetime: Let $Σ$ be a boundary component of some compact, time-symmetric, spacelike hypersurface $Ω$ in a time-oriented spacetime $N$ satisfying the dominant energy condition. Suppose the induced me…

We provide a direct proof for the positivity of Chen-Nester-Tung quasi-local energy with analytic reference in spherical symmetry. A hoop-type theorem for this energy is also established. Finally, the relation between Chen-Nester-Tung and Brown-York quasi-local energies will be discussed.

We analyse the definition of quasi-local energy in GR based on a Hamiltonian analysis of the Einstein-Hilbert action initiated by Brown-York. The role of the constraint equations, in particular the Hamiltonian constraint on the timelike boundary, neglected in previous studies, is emphasized here. We argue that a consis…

The X-ADM mass is shown to be equivalent to the ADM mass, proving the X-positive mass theorem in all dimensions.

The paper studies nonlinear mass concepts in 3-manifolds with nonnegative scalar curvature.