Time Flat Curves and Surfaces, Geometric Flows, and the Penrose Conjecture

摘要奖：DMS 1406396，主要研究者：Hubert L.这个项目旨在继续推进我们对爱因斯坦广义相对论的理解。 这个理论是我们理解重力的基础，并在日常使用中得到应用，例如全球定位系统（GPS）技术，现在大多数智能手机都有。 广义相对论还统一了空间、时间、质量和能量的概念，并为理解超大质量黑洞和宇宙历史上最具能量的事件大爆炸提供了框架。 更具体地说，该项目的重点是更好地理解广义相对论中的质量概念，无论是与黑洞还是定义爆炸超新星的质量有关。 由于几何分析是精确表述广义相对论所需要的数学领域，因此本课题将利用几何分析的许多技术来研究这些问题。具体而言，本课题主要有两个研究方向。第一个是出于理解时空中表面的几何和质量的愿望。 Jeff Jauregui，Marc Mars和PI发现，在一种新的几何流（称为“均匀面积扩展时间平坦流”）下，表面的霍金质量不减。关键的想法是，如果表面不是“时间平坦”的，霍金质量可以高估表面内部区域的质量，我们精确地定义了这一点。 时间平坦面是包含在静态时空的恒定时间片中的面的概念的推广。均匀面积扩展的时间平坦流，这本身就很有趣，它以恒定的速度增长表面的面积形式，同时保持表面作为一个整体，时间平坦。 这个流的存在性、唯一性和渐近性是在各种背景下研究的非常重要的问题。 第二个研究方向，与第一个相关，是理解黑洞的质量，因为它们与彭罗斯猜想有关，自1973年以来一直开放。 虽然PI在1999年证明了黎曼彭罗斯猜想对于任何数量的黑洞，这是时空的超曲面具有零第二基本形式的情况，但与时空的任何切片有关的一般情况仍然是开放的。虽然这些问题可以提出的语言的物理学，他们也是重要的声明有关标量和里奇曲率，最小曲面和平均曲率，等周区域，连接上的丛，子流形几何，几何流，和几何偏微分方程。

AbstractAward: DMS 1406396, Principal Investigator: Hubert L. BrayThis project aims to continue to advance our understanding of the implications of Einstein's theory of general relativity. This theory is at the foundation of our understanding of gravity and has applications in everyday use, such as global positioning system (GPS) technology, now present in most smart phones. General relativity also unifies the notions of space, time, mass, and energy and provides the framework for understanding supermassive black holes and the Big Bang, the most energetic event in the history of the universe. More specifically, this project focuses on finding a better understanding of the notion of mass in general relativity, whether it relates to a black hole or to defining the mass of an exploding supernova. Since geometric analysis is the field of mathematics required to precisely state general relativity, this project will use many techniques from geometric analysis to study these questions.Even more specifically, this project has two main research directions. The first is motivated by the desire to understand the geometry and mass of surfaces in spacetimes. Jeff Jauregui, Marc Mars, and the PI have found that the Hawking mass of a surface is nondecreasing under a new geometric flow called "uniformly area expanding time flat flow." The key idea is that the Hawking mass can overestimate the mass of a region inside a surface if the surface is not "time flat," which we define precisely. A time flat surface is a generalization of the concept of a surface contained in a constant time slice of a static spacetime. Uniformly area expanding time flat flow, which is interesting by itself, grows the area form of the surface at a constant rate while keeping the surface, as a whole, time flat. Existence, uniqueness, and asymptotics of this flow are very important questions to study in a variety of contexts. The second research direction, which is related to the first, is to understand the mass of black holes as they relate to the Penrose conjecture, open since 1973. While the PI proved the Riemannian Penrose conjecture for any number of black holes in 1999, which is the case when the hypersurface of a spacetime has zero second fundamental form, the general case pertaining to any slice of a spacetime is still open. While these questions can be posed in the language of physics, they are also important statements relating to scalar and Ricci curvature, minimal surfaces and mean curvature, isoperimetric regions, connections on bundles, submanifold geometry, geometric flows, and geometric partial differential equations.