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

时间平坦曲线和曲面、几何流和彭罗斯猜想

• 批准号: 1406396
• 负责人: Hubert Bray
• 金额: $ 21.4万
• 依托单位: Duke University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-08-01 至 2018-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1406396&HistoricalAwards=false
• 关键词: Time; Flat; Curves; Surfaces; Geometric

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.

摘要奖：DMS 1406396，首席研究员：休伯特·L·布莱尔这个项目旨在继续推进我们对爱因斯坦广义相对论含义的理解。这一理论是我们理解重力的基础，并在日常使用中得到应用，例如现在大多数智能手机中都存在的全球定位系统(GPS)技术。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大爆炸提供了框架，大爆炸是宇宙历史上能量最高的事件。更具体地说，这个项目的重点是更好地理解广义相对论中的质量概念，无论它是与黑洞有关，还是与定义爆炸中的超新星的质量有关。由于几何分析是精确表述广义相对论所需要的数学领域，本项目将使用几何分析中的许多技术来研究这些问题。更具体地说，本项目有两个主要研究方向。第一个动机是想要了解时空中表面的几何形状和质量。Jeff Jauregui、Marc Mars和PI发现，在一种名为“均匀面积扩展时间平坦流”的新几何流下，表面的霍金质量是不减的。其关键思想是，如果一个表面不是我们精确定义的“时间平坦的”，霍金质量可能会高估该表面内部区域的质量。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。均匀的面积扩展时间平坦流本身就很有趣，它以恒定的速度增长曲面的面积形式，同时保持曲面作为一个整体的时间平坦。这种流动的存在性、唯一性和渐近性是在各种背景下需要研究的非常重要的问题。与第一个研究方向相关的第二个研究方向是了解黑洞的质量，因为它们与1973年以来公开的彭罗斯猜想有关。虽然PI在1999年证明了任意数量的黑洞的黎曼-彭罗斯猜想，这是当时空的超曲面具有零秒基本形式时的情况，但关于任何时空切片的一般情况仍然是开放的。虽然这些问题可以用物理学的语言提出，但它们也是与标量曲率和Ricci曲率、极小曲面和平均曲率、等周域、丛上的联系、子流形几何、几何流动和几何偏微分方程有关的重要陈述。