On the Geometry of Static Spacetimes

Global and uniqueness properties of stationary and static spacetimes with outer trapped surfaces

Global properties of maximal future Cauchy developments of stationary, m-dimensional asymptotically flat initial data with an outer trapped boundary are analyzed. We prove that, whenever the matter model is well posed and satisfies the null energy condition, the future Cauchy development of the data is a black hole spacetime. More specifically, we show that the future Killing development of the exterior of a sufficiently large sphere in the initial data set can be isometrically embedded in the maximal Cauchy development of the data. In the static setting we prove, by working directly on the in…

任意维度（电）静态时空中等势光子面的研究

On equipotential photon surfaces in (electro-)static spacetimes of arbitrary dimension

本文在任意维度静态渐近平坦时空中刻画了球对称时空中的光子面，给出等势光子面附近几何的新见解及不依赖场方程的光子球刻画，并证明具有等势次极值光子面或非简并黑洞视界内边界的静电电真空时空必等距于相应质量与电荷的次极值 Reissner–Nordström 时空。

静态时空上的几乎梯度里奇孤子

Almost Gradient Ricci Solitons on Static Spacetime

本文研究静态时空上几乎梯度里奇孤子的几何性质，确定其存在条件，证明在真空静态时空下退化为稳态孤子，并探讨膨胀孤子满足薛定谔方程、能量条件及底流形为爱因斯坦流形等结果。

Uniqueness of the AdS Spacetime among Static Vacua with prescribed Null infinity

We prove that an $(n+1)$-dimensional spin static vacuum with negative cosmological constant whose null infinity has a boundary admitting a non-trivial Killing spinor field is the AdS spacetime. As a consequence, we generalize previous uniqueness results by X. Wang \cite{Wa2} and by Chru{ś}ciel-Herzlich \cite{CH} and introduce, for this class of spin static vacua, some Lorentzian manifolds which are prohibited as null infinities.

Spacelike hypersurfaces in standard static spacetimes

In this work we study spacelike hypersurfaces immersed in spatially open standard static spacetimes with complete spacelike slices. Under appropriate lower bounds on the Ricci curvature of the spacetime in directions tangent to the slices, we prove that every complete CMC hypersurface having either bounded hyperbolic angle or bounded height is maximal. Our conclusions follow from general mean curvature estimates for spacelike hypersurfaces. In case where the spacetime is a Lorentzian product with spatial factor of nonnegative Ricci curvature and sectional curvatures bounded below, we also show…

一类静态时空中零能量条件对全脐超曲面的影响

On effects of the null energy condition on totally umbilic hypersurfaces in a class of static spacetimes

本文研究一类静态时空中零能量条件对全脐超曲面的影响，分别在类空和类时情形下刻画了常时空平均曲率嵌入曲面与常脐性因子光子曲面。

On causality and closed geodesics of compact Lorentzian manifolds and static spacetimes

Some results related to the causality of compact Lorentzian manifolds are proven: (1) any compact Lorentzian manifold which admits a timelike conformal vector field is totally vicious, and (2) a compact Lorentzian manifold covered regularly by a globally hyperbolic spacetime admits a timelike closed geodesic, if some natural topological assumptions (fulfilled, for example, if one of the conjugacy classes of deck transformations containing a closed timelike curve is finite) hold. As a consequence, any compact Lorentzian manifold conformal to a static spacetime is geodesically connected by causa…

Uniqueness of the de Sitter spacetime among static vacua with positive cosmological constant

We prove that, among all (n + 1)-dimensional spin static vacua with positive cosmological constant, the de Sitter spacetime is characterized by the fact that its spatial Killing hori-zons have minimal modes for the Dirac operator. As a consequence, the de Sitter spacetime is the only vacuum of this type for which the induced metric tensor on some of its Killing horizons is at least equal to that of a round (n -- 1)-sphere. This extends unique-ness theorems shown by Boucher-Gibbons-Horowitz and Chruciel to more general horizon metrics and to the non-single horizon case.

Quasi-local mass and isometric embedding with reference to a static spacetime

The mathematical theory of isometric embedding is applied to study the notion of quasilocal mass in general relativity. In particular, I shall report some recent progress of quasilocal mass with reference to a cosmological spacetime, such as the de Sitter or the Anti-de Sitter spacetime, or a blackhole spacetime, such as the Schwarzschild spacetime. This article is based on joint work with Po-Ning Chen, Ye-Kai Wang, and Shing-Tung Yau.

On the Geometry and Mass of Static, Asymptotically AdS Spacetimes, and the Uniqueness of the AdS Soliton

We prove two theorems, announced in hep-th/0108170, for static spacetimes that solve Einstein's equation with negative cosmological constant. The first is a general structure theorem for spacetimes obeying a certain convexity condition near infinity, analogous to the structure theorems of Cheeger and Gromoll for manifolds of non-negative Ricci curvature. For spacetimes with Ricci-flat conformal boundary, the convexity condition is associated with negative mass. The second theorem is a uniqueness theorem for the negative mass AdS soliton spacetime. This result lends support to the new positive …

An imbedding of spacetimes

It is shown that any two-dimensional spacetimes with compact Cauchy surfaces can be causally isomorphically imbedded into the two-dimensional Einstein's static universe. Also, it is shown that any two-dimensional globally hyperbolic spacetimes are conformally equivalent to a subset of the two-dimensional Einstein's static universe.

具有Λ<0的静态真空时空作为Ricci-谐波流的吸引子

Static Vacuum Spacetimes with $Λ<0$ as Attractors of the Ricci-Harmonic Flow

研究了Λ<0的静态渐近超几何解的动态稳定性，证明了正质量定理与Ricci-谐波流的稳定性之间的关系。

Nonassociative geometry: Towards discrete structure of spacetime

In the framework of nonassociative geometry (hep-th/0003238) a unified description of continuum and discrete spacetime is proposed. In our approach at the Planck scales the spacetime is described as a so-called "diodular discrete structure" which at large spacetime scales `looks like' a differentiable manifold. After a brief review of foundations of nonassociative geometry,we discuss the nonassociative smooth and discrete de Sitter spacetimes.

Nonassociative geometry: Friedmann-Robertson-Walker spacetime

In (Phys. Rev. D 62, 081501, 2000) we proposed a unified approach to description of continuous and discrete spacetime based on nonassociative geometry and described nonassociative smooth and discrete de Sitter models. In our paper we give the description of nonassociative Friedmann-Robertson-Walker spacetime.