Yuguang Shi, Luen-Fai Tam
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.
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A compact approach to the positivity of Brown-York's mass and its relation with the Min-Oo conjecture, Yau's Problem \#100 and rigidity of hypersurfaces
本文提出一种紧致方法探讨Brown-York质量的正性，并揭示其与Min-Oo猜想、Yau问题#100及超曲面刚性的内在联系。
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.
Compact approach to the positivity of Brown-York mass and rigidity of manifolds with mean-convex boundaries in flat and spherical contexts
本文利用旋量证明与 ADM 质量正定性，为 Brown-York 质量正定性提供了紧致方法，并得出 Dirac 算子特征值下界及平坦与球面空间中平均凸流形的刚性结果。
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.
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 positive mass theorem for Brown-York mass. With the aid of positive mass theorem, we investigate the geometric structure of the so-called CPE manifolds, which provides us further understanding of Besse's conjecture. As a related topic, we also have a discussion of corresponding r…
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 eigenvalue. This inequality implies that for a compact n-dimensional manifold with boundary, its first eigenvalue is no less that n, if its scalar curvature is at least n(n-1) with appropriate boundary conditions posed, where equality holds if and only the manifold is isometric to the…
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 deformations which either increase or decrease scalar curvature. We found local conformal rigidity phenomena occur in both cases for small domains and as for manifolds with nonpositive scalar curvature it is even more rigid in particular. On the other hand, such deformations exist fo…
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 also study expansions of the Brown-York mass and the Hawking mass of geodesic spheres with center at a fixed point $p$ of a three manifold. Some geometric consequences will be derived.
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 a compact three manifold with smooth boundary. From these estimates, we generalize some previous results of the authors.
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 formula expressing the asymptotic limit of this quasi-local mass as a bulk integral which manifestly has the right sign under the above assumptions. In the spirit of recent work by Hijazi, Montiel and Raulot, we also provide another proof of this result which uses the theory of b…
Intrinsic Brown--York Type Mass at Infinity in Four Dimensions
针对四维渐近平坦流形中的闭超曲面，定义内蕴参考平均曲率，导出边界项收敛至ADM质量并含形状修正的展开式，对近球面修正项在自然衰减条件下消失。
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 consists of 2-spheres, we demonstrate that the finiteness of the supremum follows from the previous work of Shi-Tam and Wang-Yau on the quasi-local mass problem in general relativity. In turn, we define a variational analog of Brown-York quasi-local mass without assuming that the bound…
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 by the study of Brown-York mass. We prove that critical points of this variation problem are exactly static metrics. In the second part, we derive a derivative formula for the Brown-York mass of a smooth family of closed 2 dimensional surfaces evolving in an ambient three dimensi…
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.
Motivated by the important work of Brown adn York on quasilocal energy, we propose definitions of quasilocal energy and momentum surface energy of a spacelike 2-surface with positive intrinsic curvature in a spacetime. We show that the quasilocal energy of the boundary of a compact spacelike hypersurface which satisfies the local energy condition is strictly positive unless the spacetime is flat along the spacelike hypersurface.
Compact Proof of the Positivity of Quasi-Local Masses for a class of Initial Data
本文针对一类初值数据，通过Jang形变和共形约化至零标量曲率，利用Brown-York质量的第二变分，在紧致填充内证明了Wang-Yau质量的严格正性，无需渐近平坦延拓或正质量定理，部分回答了Schoen的问题。
We present a quasi-local version of the stability of the positive mass theorem. We work with the Brown--York quasi-local mass as it possesses positivity and rigidity properties, and therefore the stability of this rigidity statement can be studied. Specifically, we ask if the Brown--York mass of the boundary of some compact manifold is close to zero, must the manifold be close to a Euclidean domain in some sense? Here we consider a class of compact $n$-manifolds with boundary that can be realized as graphs in $\mathbb{R}^{n+1}$, and establish the following. If the Brown--York mass of the bound…
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, the topological sphere can be isometrically into Euclidean 3-space and mean curvature of the initial date set does not change sign. The positivity holds if we further assume the image of the topological sphere in Euclidean 3-space has nonnegative mean curvature.
A Positive Mass Type Theorem for a Singular Toroidal Static Model
本文利用逆平均曲率流和Shi-Tam型构造，为具有平坦环面切片的静态空间模型的三维流形建立了正质量型定理及Brown-York型不等式，并探讨了推广障碍及负质量与内部奇异性和拓扑的关系。
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.
In this article, we study the geometry of compact quasi-Einstein manifolds with boundary. We establish sharp boundary estimates for compact quasi-Einstein manifolds with boundary that improve some previous results. Moreover, we obtain a characterization theorem for such manifolds in terms of the surface gravity of the boundary components, which leads to a new sharp geometric inequality. In addition, we prove a boundary estimate for compact quasi-Einstein manifolds with (possibly disconnected) boundary in terms of the Brown-York mass.
Well-posed geometric boundary data in General Relativity, I: Dirichlet boundary data
本文证明了在有限类时边界上具有Dirichlet边界数据的真空爱因斯坦方程初边值问题的局部时间适定性，前提是边界的Brown-York应力张量是与边界上诱导洛伦兹度量具有相同符号（整体符号除外）的洛伦兹度量。
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 scali…
Stability of Positive Mass Theorem for Static Quasi-Local Energy of Compact (Locally) Hyperbolic Graphical Manifolds
该研究针对（局部）双曲静态空间上的紧致带边界图流形，证明了其外边界静态拟局部Brown-York能量的正质量定理在Federer–Fleming平坦距离下的稳定性。
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 complete manifolds.The method is based on analyzing Hawking mass and inverse mean curvature flow.
Boundary Behavior of Compact Manifolds With Scalar Curvature Lower Bounds and Static Quasi-Local Mass of Tori
本文将 Shi-Tam 边界平均曲率不等式推广至负曲率背景中的环面，借助第二边界的捕获条件证明静态 Brown-York 质量正性及 Penrose 型不等式。
Foliations by critical surfaces of the Hawking energy in asymptotically flat initial data sets
本文研究了霍金能量的临界面在渐近Schwarzschild初值数据集中的叶状结构，证明了其存在性和唯一性，并探讨了其坐标中心特性及大球极限下的能量行为。
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 geometric invariants of a family of nearly round surfaces which approach to infinity of the manifold.
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 the ADM mass. They may be interpreted as localized versions of these inequalities, and are also closely tied to the conjectured Bekenstein bounds for entropy of macroscopic bodies. In addition, we give a new proof of the positivity property for the Wang-Yau mass which is used to r…
Some Remarks on Wang-Yau Quasi-Local Mass
本文探讨了王-亚乌准局部质量的定义及其与布朗-亚克和全球 ADM 定义的联系，并扩展了其正性证明以允许严格稳定的视界存在。
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 nonnegative scalar curvature and, in the process, we consider fill-ins with singular metrics, which may have independent interest. Among other things, our work yields new variational characterizations of Riemannian Schwarzschild manifolds and new comparison results for surfaces in them.
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 point (0,0) and find that it is a saddle critical point in most of the cases. Also, as a byproduct, we generalize a previous result about the coincidence of CNT quasi-local energy and Brown-York mass for Kerr-like spacetimes by Tam and the first author to general spacetimes.
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 round surfaces including large constant mean curvature spheres in asymptotically flat 3-manifolds with positive mass; it is also positive definite on small geodesics spheres, whose centers do not have vanishing curvature, in Riemannian 3-manifolds of nonnegative scalar curvature. We …
A Quasi-Local Mass
本文提出一种新的规范无关拟局部质量，证明其正性、刚性及与ADM质量的收敛关系，并给出最优等距嵌入方程。
A Positive Quasilocal Mass for Causal Variational Principles
本文针对因果变分原理极小化子证明新不等式，用于正质量定理新证明，并引入正拟局部质量与标量曲率定义，在超静态及史瓦西时空示例中阐明与ADM及Brown-York质量的联系。
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