轴对称外边缘受陷二维球面的稳定性与刚性

Rigidity of stable marginally outer trapped surfaces in initial data sets

In this article we investigate the restrictions imposed by the dominant energy condition (DEC) on the topology and conformal type of \textsl{possibly non-compact} marginally outer-trapped surfaces (thus extending Hawking's classical theorem on the topology of black holes). We first prove that an unbounded, stable marginally outer trapped surface in an initial data set $(M,g,k)$ obeying the dominant energy condition is conformally diffeomorphic to either the plane $\mathbb{C}$ or to the cylinder $\mathbb{A}$ and in the latter case infinitesimal rigidity holds. As a corollary, when the DEC holds…

Stability and rigidity of extremal surfaces in Riemannian geometry and General Relativity

We examine some common features of minimal surfaces, nonzero constant mean curvature surfaces and marginally outer trapped surfaces, concerning their stability and rigidity, and consider some applications to Riemannian geometry and general relativity.

多面体的初始数据集刚性结果

Initial data set rigidity results for polyhedra

本文利用旋量证明了多面体初始数据集的二面角型刚性，将时空正质量定理的刚性分析扩展至扭曲旋量丛设定，揭示了时空正质量定理、二面角刚性与带毛细管的边缘被困曲面之间的联系。

Initial data rigidity results

We present several rigidity results for initial data sets motivated by the positive mass theorem. An important step in our proofs here is to establish conditions that ensure that a marginally outer trapped surface is "weakly outermost". A rigidity result for Riemannian manifolds with a lower bound on their scalar curvature is included as a special case.

带电初始数据集的若干刚性结果

Some rigidity results for charged initial data sets

本文研究带电初始数据集中边缘外 trapped 表面的刚性结果，扩展了作者先前工作，关注包含电荷的初始数据集。

Rigidity of outermost MOTS - the initial data version

In [5], a rigidity result was obtained for outermost marginally outer trapped surfaces (MOTSs) that do not admit metrics of positive scalar curvature. This allowed one to treat the "borderline case" in the author's work with R. Schoen concerning the topology of higher dimensional black holes [8]. The proof of this rigidity result involved bending the initial data manifold in the vicinity of the MOTS within the ambient spacetime. In this note we show how to circumvent this step, and thereby obtain a pure initial data version of this rigidity result and its consequence concerning the topology of…

Rigidity of marginally outer trapped 2-spheres

In a recent work, Galloway [9] proved a local foliation theorem by MOTSs for a 3-dimensional initial data set $(M,g,K)$ with mean curvature $τ\le0$ in a 4-dimensional spacetime $(\overline M,\overline g)$ when (under suitable assumptions) $M$ has a stable spherical MOTS $Σ$ which achieves an upper bound for the area. He proved that each leaf is a round 2-sphere with the same constant Gaussian curvature. Here, we improve his result by dropping the hypothesis on the mean curvature of $M$ and showing that each leaf is a minimal surface. We show that in an outer neighborhood $U$ of $Σ$, the metric…

带边界初始数据集中自由边界超曲面的刚性结果

Rigidity results for free boundary hypersurfaces in initial data sets with boundary

本文将两篇文献中的局部分裂定理和自由边界超曲面结果，分别推广到带边界流形和带边界初始数据集中的自由边界MOTS情形，得到带边界初始数据集中紧致自由边界超曲面的若干刚性结果。

面积-电荷不等式与带电初始数据集中的局部刚性

Area-charge inequality and local rigidity in charged initial data sets

本文研究爱因斯坦-麦克斯韦方程中面积-电荷不等式等号成立时的几何刚性，证明在适当条件下饱和该不等式会强制曲面邻域具有刚性结构，并建立了时间对称与非时间对称两种情形的刚性定理。

A Penrose-Type Inequality with Angular Momentum and Charge for Axisymmetric Initial Data

A lower bound for the ADM mass is established in terms of angular momentum, charge, and horizon area in the context of maximal, axisymmetric initial data for the Einstein-Maxwell equations which satisfy the weak energy condition. If, on the horizon, the given data agree to a certain extent with the associated model Kerr-Newman data, then the inequality reduces to the conjectured Penrose inequality with angular momentum and charge. In addition, a rigidity statement is also proven whereby equality is achieved if and only if the data set arises from the canonical slice of a Kerr-Newman spacetime.

Geometry of non-compact minimal and marginally outer-trapped surfaces in asymptotically flat manifolds

In this article we extend several foundational results of the theory of complete minimal surfaces of finite index in the Euclidean space to minimal surfaces in asymptotically flat manifolds and, more generally, to marginally outer-trapped surfaces in initial data sets of General Relativity. We show that if an asymptotically flat 3-manifold (M,g) of nonnegative scalar curvature contains a non-compact, properly embedded minimal surface which is stable and has quadratic area growth, then it is isometric to the flat R^{3}. This implies, for instance, that in presence of a positive ADM mass any seq…

Rigidity of free boundary MOTS

The aim of this work is to present an initial data version of Hawking's theorem on the topology of back hole spacetimes in the context of manifolds with boundary. More precisely, we generalize the results of G. J. Galloway and R. Schoen [13] and G. J. Galloway [11, 12] by proving that a compact free boundary stable marginally outer trapped surface (MOTS) $Σ$ in an initial data set with boundary satisfying natural dominant energy conditions (DEC) is of positive Yamabe type, i.e. $Σ$ admits a metric of positive scalar curvature with minimal boundary, provided $Σ$ is outermost. To do so, we prove…

Relating Mass to Angular Momentum and Charge in 5-Dimensional Minimal Supergravity

We prove a mass-angular momentum-charge inequality for a broad class of maximal, asymptotically flat, bi-axisymmetric initial data within the context of five-dimensional minimal supergravity. We further show that the charged Myers-Perry black hole initial data are the unique minimizers. In addition, we establish a rigidity statement for the relevant BPS bound, and give a variational characterization of BMPV black holes.

On extreme Kerr-throats and zero temperature black-holes

Recently it was shown that the area A and the angular momentum J of any apparent horizon on a maximal, axisymmetric and asymptotically flat Cauchy hyper-surface of a vacuum space-time satisfy necessarily the universal inequality A >= 8 pi |J|. We show here that the equality A=8 pi |J| is never attained. As equality is reached (on globally different data sets) when and only when the surface is an extreme Kerr-throat sphere which has zero "temperature", then our statement could be rephrased following this thermodynamic heuristic as the non-existence of apparent horizons of zero temperature. We s…

Spacetime Harmonic Functions and the Mass of 3-Dimensional Asymptotically Flat Initial Data for the Einstein Equations

We give a lower bound for the Lorentz length of the ADM energy-momentum vector (ADM mass) of 3-dimensional asymptotically flat initial data sets for the Einstein equations. The bound is given in terms of linear growth `spacetime harmonic functions' in addition to the energy-momentum density of matter fields, and is valid regardless of whether the dominant energy condition holds or whether the data possess a boundary. A corollary of this result is a new proof of the spacetime positive mass theorem for complete initial data or those with weakly trapped surface boundary, and includes the rigidity…