On the Center of Mass of Asymptotically Hyperbolic Initial Data Sets

We define the (total) center of mass for suitably asymptotically hyperbolic time-slices of asymptotically anti-de Sitter spacetimes in general relativity. We do so in analogy to the picture that has been consolidated for the (total) center of mass of suitably asymptotically Euclidean time-slices of asymptotically Minkowskian spacetimes (isolated systems). In particular, we unite—an altered version of—the approach based on Hamiltonian charges with an approach based on CMC-foliations near infinity. The newly defined center of mass transforms appropriately under changes of the asymptotic coordinates and evolves in the direction of an appropriately defined linear momentum under the Einstein evolution equations.     

Rigidity and Positivity of Mass for Asymptotically Hyperbolic Manifolds

The Witten spinorial argument has been adapted in several works over the years to prove positivity of mass in the asymptotically AdS and asymptotically hyperbolic settings in arbitrary dimensions. In this paper we prove a scalar curvature rigidity result and a positive mass theorem for asymptotically hyperbolic manifolds that do not require a spin assumption. The positive mass theorem is reduced to the rigidity case by a deformation construction near the conformal boundary. The proof of the rigidity result is based on a study of minimizers of the BPS brane action. Submitted: March 16, 2007. Accepted: June 14, 2007.     

The Brown–York Mass of Revolution Surfaces in Asymptotically Schwarzschild 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.     

A Positive Mass Theorem on Asymptotically Hyperbolic Manifolds with Corners along a Hypersurface

In this paper we take an approach similar to that in [13] to establish a positive mass theorem for spin asymptotically hyperbolic manifolds admitting corners along a hypersurface. The main analysis uses an integral representation of a solution to a perturbed eigenfunction equation to obtain an asymptotic expansion of the solution in the right order. This allows us to understand the change of the mass aspect of a conformal change of asymptotically hyperbolic metrics. Vincent Bonini: The first named author supported by MSRI Postdoctoral Fellowship. Jie Qing: The second named author supported partially by NSF grant DMS 0402294. Submitted: April 6, 2007. Accepted: September 24, 2007.     

A Generalized Mass Involving Higher Order Symmetric Functions of the Curvature Tensor

We define a generalized mass for asymptotically flat manifolds using some higher order symmetric function of the curvature tensor. This mass is non-negative when the manifold is locally conformally flat and the σ _k curvature vanishes at infinity. In addition, with the above assumptions, if the mass is zero, then, near infinity, the manifold is isometric to a Euclidean end.     

Regularity of inverse mean curvature flow in asymptotically hyperbolic manifolds with dimension 3

By using the nice behavior of the Hawking mass of the slices of a weak solution of inverse mean curvature flow in three-dimensional asymptotically hyperbolic manifolds, we are able to show that each slice of the flow is star-shaped after a long time, and then we get the regularity of the weak solution of inverse mean curvature flow in asymptotically hyperbolic manifolds. As an application, we prove that the limit of the Hawking mass of the slices of a weak solution of inverse mean curvature flow with any connected C~2-smooth surface as initial data in asymptotically anti-de Sitter-Schwarzschild manifolds with positive mass is greater than or equal to the total mass, which is completely different from the situation in the asymptotically flat case.     

Time Flat Surfaces and the Monotonicity of the Spacetime Hawking Mass II

In this sequel paper, we give a shorter, second proof of the monotonicity of the Hawking mass for time flat surfaces under spacelike uniformly area expanding flows in spacetimes that satisfy the dominant energy condition. We also include a third proof which builds on a known formula and describe a class of sufficient conditions of divergence type for the monotonicity of the Hawking mass. These flows of surfaces may have connections to the problem in general relativity of bounding the total mass of a spacetime from below by the quasi-local mass of spacelike 2-surfaces in the spacetime.     

Gauss-Bonnet-Chern mass and Alexandrov-Fenchel inequality

This is a survey about our recent works on the Gauss-Bonnet-Chern (GBC) mass for asymptotically flat and asymptotically hyperbolic manifolds. We first introduce the GBC mass, a higher order mass, for asymptotically flat and for asymptotically hyperbolic manifolds, respectively, by using a higher order scalar curvature. Then we prove its positivity and the Penrose inequality for graphical manifolds. One of the crucial steps in the proof of the Penrose inequality is the use of an Alexandrov-Fenchel inequality, which is a classical inequality in the Euclidean space. In the hyperbolic space, we have established this new Alexandrov-Fenchel inequality. We also have a similar work for asymptotically locally hyperbolic manifolds. At the end, we discuss the relation between the GBC mass and Chern’s magic form.     

On the capacity of surfaces in manifolds with nonnegative scalar curvature

Given a surface in an asymptotically flat 3-manifold with nonnegative scalar curvature, we derive an upper bound for the capacity of the surface in terms of the area of the surface and the Willmore functional of the surface. The capacity of a surface is defined to be the energy of the harmonic function which equals 0 on the surface and goes to 1 at ∞. Even in the special case of ℝ³, this is a new estimate. More generally, equality holds precisely for a spherically symmetric sphere in a spatial Schwarzschild 3-manifold. As applications, we obtain inequalities relating the capacity of the surface to the Hawking mass of the surface and the total mass of the asymptotically flat manifold.     

On the uniqueness of the foliation of spheres of constant mean curvature in asymptotically flat 3-manifolds

In this note we study constant mean curvature surfaces in asymptotically flat 3-manifolds. We prove that, outside a given compact subset in an asymptotically flat 3-manifold with positive mass, stable spheres of given constant mean curvature are unique. Therefore we are able to conclude that the foliation of stable spheres of constant mean curvature in an asymptotically flat 3-manifold with positive mass outside a given compact subset is unique.

     

Conformal positive mass theorems for asymptotically flat manifolds with inner boundary

Inspired by Witten's insightful spinor proof of the positive mass theorem, in this paper, we use the spinor method to derive higher dimensional type conformal positive mass theorems on asymptotically flat spin manifolds with inner boundary, which states that under a condition about the plus (minus) relation between the scalar curvatures of the original and the conformal metrics in addition with some boundary condition, we will get the associated positivity of their ADM masses. The rigidity part of the plus part is used in the proof of black hole uniqueness theorems. They are related with quasi-local mass and the spectrum of Dirac operator.     

Banach空间中几乎渐近非扩张型映象的不动点的迭代逼近

在Banach空间中引入了一类新的几乎渐近非扩张型映象,概括了Banach空间中若干熟知的非线性的Lipschitz映象类与非Lipschitz映象类成特例;例如,熟知的非扩张映象类,渐近非扩张映象类与渐近非扩张型映象类.考虑了用于逼近几乎渐近非扩张型映象不动点的带误差的修改了的Ishikawa迭代序列的收敛性问题.关于Banach空间范数的S.S.Chang的不等式与H.K.Xu的不等式皆被用于做精确不动点与近似不动点间的误差估计.而且,张石生教授用于做带误差的修改了的Ishikawa迭代序列收敛性分析的方法(应用数学和力学,2001,22(1):23-31)被推广到几乎渐近非扩张型映象的情况.给出了用于求一致凸Banach空间中几乎渐近非扩张型映象不动点的带误差的修改了的Ishikawa迭代序列的新的收敛判据.并且,由该判据,立即得到了此类映象的带误差的修改了的Mann迭代序列的新的收敛判据.上述结果统一、改进与推广了张石生教授关于用带误差的修改了的Ishikawa与Mann迭代序列来逼近渐近非扩张型映象不动点方面的结果.     

Computing Asymptotic Invariants with the Ricci Tensor on Asymptotically Flat and Asymptotically Hyperbolic Manifolds

We prove in a simple and coordinate-free way the equivalence between the classical definitions of the mass or of the center of mass of an asymptotically flat manifold and their alternative definitions depending on the Ricci tensor and conformal Killing fields. This enables us to prove an analogous statement in the asymptotically hyperbolic case.     

On the behavior of quasi-local mass at the infinity along nearly round surfaces

In this article, 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 three manifold can be approximated by some geometric invariants of a family of nearly round surfaces, which approach to infinity of the manifold.     

Asymptotic normality of discounted random series with applications in reliability and queueing

It is proved that a certain kind of randomly discounted random sums is asymptotically normal as the discount constant tends to zero. For replaceable systems with random lifetime, these sums represent the total discounted cost of policies of the age-replacement type; other applications to queueing and related areas are also indicated.     

Conformal CR Positive Mass Theorem

In this paper, we prove the following version of conformal CR positive mass theorem: Suppose that \((N, J,\theta )\) and \((N, J,\hat{\theta }=e^{2f}\theta )\) are three-dimensional asymptotically flat pseudohermitian manifolds such that their Tanaka-Webster curvatures satisfy \(e^{2f}\hat{R}-R\ge 0.\) Then the p-mass of \((N, J, \theta )\) and \((N, J, \hat{\theta })\) satisfy \( m(J, \hat{\theta })-m(J, \theta )\ge 0, \) and equality holds if and only if \(\hat{\theta }=\theta \). We also prove that the p-mass is independent of the choice of the sequence of coordinates spheres.     

Absence of Critical Mass Densities for a Vibrating Membrane

We study the dependence of the eigenvalues of a N-dimensional vibrating membrane upon variation of the mass density. We prove that the elementary symmetric functions of the eigenvalues depend real-analytically on the mass density and that such functions have no critical points with constant mass constraint. In particular, the elementary symmetric functions of the eigenvalues, hence all simple eigenvalues, have no local maxima or minima on the set of those mass densities with a prescribed total mass.     

Large Time Behavior for a Simplified 1D Model of Fluid–Solid Interaction†

Abstract

In this article we consider a simple model in one space dimension for the interaction between a fluid and a solid represented by a point mass. The fluid is governed by the viscous Burgers equation and the solid mass, which shares the velocity of the fluid, is accelerated by the difference of pressure at both sides of it. We describe the asymptotic behavior of solutions for integrable data using energy estimates and scaling techniques. We prove that the asymptotic profile of the fluid is a self-similar solution of the Burgers equation with an appropriate total mass, and we describe the parabolic trajectory of the point mass. We also prove that, asymptotically, the difference of pressure to both sides of the point mass vanishes.     

Penrose-type inequalities with a Euclidean background

The Riemannian Penrose inequality (RPI) bounds from below the ADM mass of asymptotically flat manifolds of nonnegative scalar curvature in terms of the total area of all outermost compact minimal surfaces. The general form of the RPI is currently known for manifolds of dimension up to seven. In the present work, we prove a Penrose-like inequality that is valid in all dimensions, for conformally flat manifolds. Our inequality treats the area contributions of the minimal surfaces in a more favorable way than the RPI, at the expense of using the smaller Euclidean area (rather than the intrinsic area). We give an example in which our estimate is sharper than the RPI when many minimal surfaces are present. We do not require the minimal surfaces to be outermost. We also generalize the technique to allow for metrics conformal to a scalar-flat (not necessarily Euclidean) background and prove a Penrose-type inequality without an assumption on the sign of scalar curvature. Finally, we derive a new lower bound for the ADM mass of a conformally flat, asymptotically flat manifold containing any number of zero area singularities.     

Estimating functionals of one-dimensional Gibbs states

Summary Some estimators of maximum likelihood type are constructed for estimating functionals of one-dimensional Gibbs states. We also show that those estimators are strongly consistent, asymptotically normal and asymptotically efficient.