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.

Abstractaward：DMS 1406396，首席研究员：Hubert L. Braythis项目旨在继续提高我们对爱因斯坦一般相对论含义的理解。 该理论是我们对重力的理解的基础，并且在日常使用中具有应用，例如现在存在于大多数智能手机中的全球定位系统（GPS）技术。 一般相对性也统一了时空，时间，质量和能量的概念，并为理解超大型黑洞和大爆炸提供了框架，这是宇宙历史上最有活力的事件。 更具体地说，该项目致力于找到对一般相对性质量概念的更好理解，无论是与黑洞有关还是定义爆炸超新星的质量。 由于几何分析是精确说明一般相对性所需的数学领域，因此该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，该项目具有两个主要的研究方向。第一个是出于渴望了解空间中表面的几何形状和质量的愿望。 Jeff Jauregui，Marc Mars和PI发现表面的鹰质量在新的几何流动下不稳定，称为“统一区域扩大了时间平坦的流动”。 关键的想法是，如果表面不是“时间平坦”，那么霍金质量可以高估表面内部区域的质量，我们会精确地定义。 时间平面表面是对静态恒定时间切片中表面概念的概括。均匀的区域扩展时间扁平流，这本身很有趣，它本身就以恒定的速率生长表面的面积形式，同时整体上保持表面平坦。 这种流程的存在，独特性和渐近学是在各种情况下研究的非常重要的问题。 与第一个相关的第二个研究方向是要了解与Penrose猜想相关的黑洞的质量，自1973年以来开放。尽管PI证明了1999年的Riemannian Penrose猜想对任何数量的黑洞的猜想，这是当时的零次生型的超级态度时，零是一个普遍的情况。尽管这些问题可以用物理语言提出，但它们也是与标量和RICCI曲率，最小表面和平均曲率，等值区域，捆绑包上的连接，亚曼型几何形状，几何流量和几何偏差方程有关的重要陈述。