Hubert L. Bray
In this thesis we describe how minimal surface techniques can be used to prove the Penrose inequality in general relativity for two classes of 3-manifolds. We also describe how a new volume comparison theorem involving scalar curvature for 3-manifolds follows from these same techniques.
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We propose a geometric inequality for two-dimensional spacelike surfaces in the Schwarzschild spacetime. This inequality implies the Penrose inequality for collapsing dust shells in general relativity, as proposed by Penrose and Gibbons. We prove that the inequality holds in several important cases.
The positive mass theorem is one of the fundamental results in general relativity. It states that, assuming the dominant energy condition, the total mass of an asymptotically flat spacetime is non-negative. The Penrose inequality provides a lower bound on mass by the area of the black hole and is closely related to the cosmic censorship conjecture in general relativity. In [14], Lu and Miao proved a quasi-local Penrose inequality for the quasi-local energy with reference in the Schwarzschild manifold. In this article, we prove a quasi-local Penrose inequality for the quasi-local energy with re…
In a paper \cite{P} in 1973, R. Penrose made a physical argument that the total mass of a spacetime which contains black holes with event horizons of total area $A$ should be at least $\sqrt{A/16π}$. An important special case of this physical statement translates into a very beautiful mathematical inequality in Riemannian geometry known as the Riemannian Penrose inequality. One particularly geometric aspect of this problem is the fact that apparent horizons of black holes in this setting correspond to minimal surfaces in Riemannian 3-manifolds. The Riemannian Penrose inequality was first prove…
In this summary of Habilitation Thesis, it is outlined author's 18 years research activity on mathematical physics, geometric methods in particle physics and gravity, modifications and applications (after defending his PhD thesis in 1994). Ten most relevant publications are structured conventionally into three "strategic directions": 1) nonholonomic geometric flows evolutions and exact solutions for Ricci solitons and field equations in (modified) gravity theories; 2) geometric methods in quantization of models with nonlinear dynamics and anisotropic field interactions; 3) (non) commutative ge…
Consider a compact, orientable, three dimensional Riemannian manifold with boundary with nonnegative scalar curvature. Suppose its boundary is the disjoint union of two pieces: the horizon boundary and the outer boundary, where the horizon boundary consists of the unique closed minimal surfaces in the manifold and the outer boundary is metrically a round sphere. We obtain an inequality relating the area of the horizon boundary to the area and the total mean curvature of the outer boundary. Such a manifold may be thought as a region, surrounding the outermost apparent horizons of black holes, i…
The main aim of this thesis is to study the properties of trapped surfaces in spacetimes with symmetries and their possible relation with the theory of black holes. We will concetrate specially on one aspect of this possible equivalence, namely whether the static black hole uniqueness theorems extend to static spacetimes containing marginally outer trapped surfaces. The principal result of this thesis states that this question has an affirmative answer, under suitable not global-in-time conditions on the spacetime. Furthermore, in order to solve this question, we will obtain several results wh…
As early as 1972, Penrose - in a purely formal way - introduced a "discontinuous coordinate transformation", which relates a continuous representation of the metric of impulsive pp-waves to a discontinuous one. On the basis of the invertibility concept for generalized functions developed recently by the first author, we show that this discontinuous coordinate transformation indeed represents an invertible generalized function in the appropriate sense.
Three-Dimensional Almost Contact Metric Manifolds Revisited via the Newman-Penrose Formalism
本文将Newman-Penrose形式应用于三维几乎接触度量流形分析，探讨其在几何中的适用性，并分类了具有η-Einstein度量的紧致正常ACM流形。
We adapt the Newman-Penrose formalism in general relativity to the setting of three-dimensional Riemannian geometry, and prove the following results. Given a Riemannian 3-manifold without boundary and a smooth unit vector field ${\boldsymbol k}$ with geodesic flow, if an integral curve of ${\boldsymbol k}$ is hypersurface-orthogonal at a point, then it is so at every point along that curve. Furthermore, if ${\boldsymbol k}$ is complete, hypersurface-orthogonal, and satisfies $\text{Ric}({\boldsymbol k},{\boldsymbol k}) \geq 0$, then its divergence must be nonnegative. As an application, we sho…
We prove the Riemannian Penrose conjecture, an important case of a conjecture made by Roger Penrose in 1973, by defining a new flow of metrics. This flow of metrics stays inside the class of asymptotically flat Riemannian 3-manifolds with nonnegative scalar curvature which contain minimal spheres. In particular, if we consider a Riemannian 3-manifold as a totally geodesic submanifold of a space-time in the context of general relativity, then outermost minimal spheres with total area $A$ correspond to apparent horizons of black holes contributing a mass $\sqrt{A/16π}$, scalar curvature correspo…
The Penrose inequality gives a lower bound for the total mass of a spacetime in terms of the area of suitable surfaces that represent black holes. Its validity is supported by the cosmic censorship conjecture and therefore its proof (or disproof) is an important problem in relation with gravitational collapse. The Penrose inequality is a very challenging problem in mathematical relativity and it has received continuous attention since its formulation by Penrose in the early seventies. Important breakthroughs have been made in the last decade or so, with the complete resolution of the so-called…
Volume Singularities in General Relativity
本文提出广义相对论中体积奇点的新概念，证明霍金定理的奇点也是体积奇点，并指出黑洞中体积奇点总被事件视界隐藏，关联彭罗斯宇宙监督猜想。
The Penrose inequality in extrinsic geometry
本文解决了G. Huisken的外蕴几何彭罗斯不等式猜想，证明三维渐近平坦支撑曲面的外质量下界，刻画等号情形的半悬链面结构，并通过极小毛细曲面的非递减量完成证明，为悬链面提供新的几何刻画。
Displacement convexity of Boltzmann's entropy characterizes the strong energy condition from general relativity
本文将黎曼流形中里奇曲率下界与熵的测地凸性之间的对应关系推广到洛伦兹时空，证明 Hawking-Penrose 强能量条件等价于 Boltzmann-Shannon 熵在洛伦兹距离分数次幂诱导的概率测度空间中的测地凸性，为在非光滑时空中研究强能量条件奠定基础。
The central object of study of this thesis is inverse mean curvature vector flow of two-dimensional surfaces in four-dimensional spacetimes. Being a system of forward-backward parabolic PDEs, inverse mean curvature vector flow equation lacks a general existence theory. Our main contribution is proving that there exist infinitely many spacetimes, not necessarily spherically symmetric or static, that admit smooth global solutions to inverse mean curvature vector flow. Prior to our work, such solutions were only known in spherically symmetric and static spacetimes. The technique used in this thes…
We identify a condition on spacelike 2-surfaces in a spacetime that is relevant to understanding the concept of mass in general relativity. We prove a formula for the variation of the spacetime Hawking mass under a uniformly area expanding flow and show that it is nonnegative for these so-called "time flat surfaces." Such flows generalize inverse mean curvature flow, which was used by Huisken and Ilmanen to prove the Riemannian Penrose inequality for one black hole. A flow of time flat surfaces may have connections to the problem in general relativity of bounding the mass of a spacetime from b…
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