Hyperbolic positive mass theorem under modified energy condition

We provide two new positive mass theorems under respective modified energy conditions allowing T00 negative on some compact set for certain modified asymptotically hyperbolic manifolds. This work is analogous to Zhang’s previous result for modified asymptotically ?at initial data sets.     

A new quasi-local mass and positivity

A new definition of quasi-local mass is proposed and its positivity is proved under certain conditions.     

A quasi-local mass for 2-spheres with negative Gauss curvature

We extend our previous definition of quasi-local mass to 2-spheres whose Gauss curvature is negative, and prove its positivity.     

A Survey of Total Positivity

This paper contains an exposition of the classical work of Schoenberg and Gantmacher, Krein on the theory of totally positive matrices. It also contains a generalization of this theory to the Lie theoretic context and an application to the recent results of Fock and Goncharov on Teichmüller theory. Leonardo da Vinci lecture held on May 28, 2007 Received: June 2007     

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.     

Gravitational radiation and the evolution of gravitational collapse in cylindrical symmetry

Using the Sparling form and a geometric construction adapted to spacetimes with a 2-dimensional isometry group, we analyse a quasi-local measure of gravitational energy. We then study the gravitational radiation through spacetime junctions in cylindrically symmetric models of gravitational collapse to singularities. The models result from the matching of collapsing dust fluids interiors with gravitational wave exteriors, given by the Einstein–Rosen type solutions. For a given choice of a frame adapted to the symmetry of the matching hypersurface, we are able to compute the total gravitational energy radiated during the collapse and state whether the gravitational radiation is incoming or outgoing, in each case. This also enables us to distinguish whether a gravitational collapse is being enhanced by the gravitational radiation.     

Positive energy theorem for (4+1)-dimensional asymptotically anti-de Sitter spacetimes

We define the total energy-momenta for(4+1)-dimensional asymptotically anti-de Sitter spacetimes,which comes from the boundary terms at infinity in the integral form of the Weitzenbck formula.Then we prove the positive energy theorem for such spacetimes,following Witten’s original argumentsfor the positive energy theorem in asymptotically flat spacetimes.     

Existence of Positive Solutions for Singular Fourth Order Coupled System with Sign Changing Nonlinear Terms

This paper deals with the existence of positive solutions to a singular fourth order coupled system with integral boundary conditions. Since the nonlinear terms $f$, $g$ may change sign or be singular at $t = 0$ or $t = 1$, the authors make a priori estimates to overcome some difficulties and apply Guo-Krasnoselskii fixed point theorem to prove the existence of solutions of the system under suitable assumptions. Finally, some examples to illustrate the main results are given.     

Positive mass theorems for high-dimensional spacetimes with black holes

We give a rigorous proof of the positive mass theorem for high-dimensional spacetimes with black holes if the spacetime contains an asymptotically flat spacelike spin hypersurface and satisfies the dominant energy condition along the hypersurface. We also weaken the spin structure on the spacelike hypersurface to spinc structure and give a modified positive mass theorem for spacetimes with black holes in dimensions 4, 5 and 6.     

非线性微分方程的正周期解

§ 1　 IntroductionIn[1 ] ,Saker and Agarwal studied the existence and uniqueness of positive periodicsolutions of the nonlinear differential equationN′(t) =-δ(t) N(t) + p(t) N(t) e- a N(t) ,(1 )whereδ(t) and p(t) are positive T-periodic functions.They proved that if p* >δ* ,then(1 ) has a unique T-periodic positive solution,wherep* =min0≤ t≤ Tp(t) ,δ* =max0≤ t≤ Tδ(t) .　　In view ofthe papermentioned above,whatcan be said aboutequation(1 ) when p* ≤δ* ?In this paper,we conside…     

QUASI-LOCAL CONJUGACY THEOREMS IN BANACH SPACES

Let f : U(x0) (?) E→F be a. C1 map and f'(X0) be the Prechet derivative of /fat X0. In local analysis of nonlinear functional analysis, implicit function theorem, inverse function theorem, local surjectivity theorem, local injectivity theorem, and the local conjugacy theorem are well known. Those theorems are established by using the properties: f'(x0) is double splitting and R(f'(x))∩N(T0 ) = {0} near X0. However, in infinite dimensional Banach spaces, f'(x0) is not always double splitting (i.e., the generalized inverse of f'(xo) does not always exist), but its bounded outer inverse of f'(x0) always exists. Only using the C1 map f and the outer inverse T0# of f'(x0), the authors obtain two quasi-local conjugacy theorems, which imply the local conjugacy theorem if X0 is a locally fine point of f. Hence the quasi-local conjugacy theorems generalize the local conjugacy theorem in Banach spaces.     

关于$5$次对称形式正性的机器判定

利用参系数多项式正实根的判别序列,给出了多变元5次对称形式在Rn 上取非负值的显示判定方法.并以此为依据,导出了一个有效的算法,能够在变元数较多时也可以使用计算机来自动判定.     

非线性分数微分方程解的若干存在性结论

研究了非线性分数微分方程解的存在性,通过考察非线性项在无穷远处的增长或者非线性项在某个有界集上的“高度”获得了若干新的存在性结论,主要工具是Schauder不动点定理和Leray-Schauder不动点定理．     

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.     

Five-Dimensional General Relativity and Kaluza–Klein Theory

In five-dimensional gravity, we consider spaces admitting a family of maximally symmetric three-dimensional subspaces. We construct five-dimensional vacuum Einstein equations and introduce the analogue of the five-dimensional mass function for these spaces. The charge conservation law for this function results in the five-dimensional analogue of the Birkhoff theorem. Hence, for the spaces under consideration, the cylindricity condition is realized dynamically. For some of the obtained metrics, the regularity condition results in the closedness of the fifth coordinate. We can then relate the period of the fifth coordinate with the value of the conserved charge. We discuss the problem of separating dynamical degrees of freedom of scalar and gravitational fields obtained when reducing the initial five-dimensional action to the four-dimensional form and the related problem of the conformal ambiguity of the four-metric gauge. The parameterization of the scalar field and the four-metric that results in a conformally invariant theory of interacting scalar and gravitational fields seems most natural.     

Positivity of polarizations of -positive maps

It is shown that polarization formulas have explicit matrix representations. This enables us to prove that polarization formulas of -positive maps between -algebras are coordinatewise positive.

     

Positivity of the two-particle Hamiltonian on a lattice

We consider a two-particle Hamiltonian on the d-dimensional lattice ℤ^d. We find a sufficient condition for the positivity of a family of operators h(k) appearing after the “separation of the center of mass” of a system of two particles depending on the values of the total quasimomentum k ∈ T^d (where T^d is a d-dimensional torus). We use the obtained result to show that the operator h(k) has positive eigenvalues for nonzero k ∈ T^d. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 153, No. 3, pp. 381–387, December, 2007.     

二阶非自治Hamilton 系统周期解的新结果

研究了二阶非自治Hamilton系统（t）-B（t）x（t）＋▽H（t,x（t））=0,在局部超二次条件下周期解的存在性问题,利用山路定理和局部环绕定理得到了新的存在性定理,改进了已有结果.     

Remark on the limit case of positive mass theorem for manifolds with inner boundary

In (Comm. Math. Phys. 188 (1997) 121–133) Herzlich proved a new positive mass theorem for Riemannian 3-manifolds (N,g) whose mean curvature of the boundary allows some positivity. In this paper we study what happens to the limit case of the theorem when, at a point of the boundary, the smallest positive eigenvalue of the Dirac operator of the boundary is strictly larger than one-half of the mean curvature (in this case the mass m(g) must be strictly positive). We prove that the mass is bounded from below by a positive constant c(g), m(g)c(g), and the equality m(g)=c(g) holds only if, outside a compact set, (N,g) is conformally flat and the scalar curvature vanishes. The constant c(g) is uniquely determined by the metric g via a Dirac-harmonic spinor.