Scalar Curvature, the Penrose Conjecture, and the Axioms of General Relativity

标量曲率、彭罗斯猜想和广义相对论公理

• 批准号: 1007063
• 负责人: Hubert Bray
• 金额: $ 32.9万
• 依托单位: Duke University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1007063&HistoricalAwards=false
• 关键词: Scalar; Curvature; Penrose; Conjecture; Axioms

First, the PI will continue his research on scalar curvature, especially on 3 manifolds. Prior results by the PI in this area include a joint work with Andre Neves in 2002 that classifies prime 3-manifolds with Yamabe invariant greater than RP^3 and a 2008 paper with Pengzi Miao that gives an upper bound on the capacity of surfaces in 3-manifolds with nonnegative scalar curvature. In 2009, the PI's joint paper with Simon Brendle, Michael Eichmair, and AndreNeves proves that A_{min}R_{min} \le 12\pi on compact 3-manifolds which contain embedded incompressible RP^2, where A_{min} is the area of the minimal RP^2 and R_{min} is the minimum value of the scalar curvature. Using Ricci flow, they show that the 3-manifold is a spherical space form in the case of equality. Second, the PI will continue to work toward a proof of the full Penrose conjecture. The PI's 2001 paper proved the Riemannian Penrose conjecture in dimension 3, improving the case of one black hole proved by Huisken and Ilmanen to any number of black holes using a different technique. Since then, the PI proved a similar type of inequality for zero area singularities in 2005 (with some additional hypotheses), the Riemannian Penrose conjecture in dimensions less than 8 in a joint work with Dan Lee in 2007, and showed that the full Penrose conjecture on Cauchy data (M^3,g,k) reduces to the Riemannian case whenever certain systems of p.d.e.s can be solved in a joint work with Marcus Khuri in 2009. These systems of p.d.e.s rely on a new identity that they proved called the Generalized Schoen-Yau identity, which they believe will be a very useful identity for a broad range of problems in mathematical relativity. Third, the PI is opening up a new research direction for himself as he examines the axioms of general relativity to see how they may be modified as little as possible to account for the widely accepted existence of dark matter.Einstein's theory of general relativity was made possible by Gauss and Riemann, both mathematicians, who developed the field of mathematics called differential geometry decades before. Since then, advances in differential geometry have played a crucial role in understanding the implications of Einstein's theory. Einstein used differential geometry to make the qualitative statement ``matter curves spacetime'' precise, thereby showing that gravity results as a consequence of this fundamental idea. By contrast, Newton's inverse square law for gravity has been shown to be false by measuring the precession of the orbit of Mercury. Hence, understanding gravity correctly would appear to require understanding the properties of curvature, currently pursued most directly by mathematicians studying geometric analysis. Black holes, predicted by general relativity and now known to exist, are fundamentally geometric objects, and have been the focus of much of the PI's efforts, resulting in theorems which yield a deeper physical insight into these fascinating phenomena. In light of this rich history of geometric analysis playing a crucial role in understanding the large scale structure of the universe, the PI is now looking to geometric motivations to try to understand the nature of dark matter. While dark matter is known to make up 23% of the mass of the universe and hence has very important gravitational effects, it is otherwise invisible. A geometric idea observed by the PI, as well as other motivations, leads to considering a real-valued scalar field as a model for dark matter, described by the Einstein Klein-Gordon equations. Astrophysicists have already observed that this model for dark matter is consistent with the flat rotation curves of galaxies. The PI is studying the idea that density waves in this scalar field dark matter produce density waves in regular matter, resulting in star formation and both bars and spiral patterns in some galaxies, an exciting possibility supported by preliminary simulations. If correct, this would suggest that while dark matter itself is invisible, its gravitational effects may be quite dramatic.

首先，PI将继续他对标量曲率的研究，特别是3流形。PI在这一领域之前的成果包括2002年与安德烈·内维斯的联合工作，该工作对山部不变量大于RP的素三维流形进行了分类^[3]，以及2008年与苗鹏子的一篇论文，该论文给出了具有非负标量曲率的三维流形中曲面容量的上限。2009年，PI与Simon Brendle、Michael Eichmair和AndreNeves的联合论文证明了包含嵌入不可压缩RP^2的紧致3-流形上的A_{min}R_{min} \le 12\pi，其中A_{min}是最小RP^2的面积，R_{min}是标量曲率的最小值。利用Ricci流，他们证明了在等式的情况下，三维流形是一个球面空间形式。第二，PI将继续致力于证明完整的彭罗斯猜想。PI在2001年的论文中证明了黎曼彭罗斯猜想在3维空间中，改进了Huisken和Ilmanen使用不同技术证明的一个黑洞到任何数量黑洞的情况。此后，PI在2005年证明了一个类似的零面积奇点不等式（加上一些额外的假设），在2007年与Dan Lee的联合工作中，黎曼彭罗斯猜想的维数小于8，并表明柯西数据上的完整彭罗斯猜想（M^3，g，k）在2009年与Marcus Khuri的联合工作中，当某些p.d.e.s系统可以被解决时，可以简化为黎曼情况。这些p.d.e.s系统依赖于他们证明的一个新恒等式，称为广义舍恩-丘恒等式，他们认为这将是一个非常有用的恒等式，适用于数学相对论中的广泛问题。第三，PI为自己开辟了一个新的研究方向，他研究了广义相对论的公理，看看如何尽可能少地修改它们，以解释被广泛接受的暗物质的存在。爱因斯坦的广义相对论是由高斯和黎曼提出的，他们都是数学家，他们在几十年前发展了称为微分几何的数学领域。从那时起，微分几何的进步在理解爱因斯坦理论的含义方面发挥了至关重要的作用。爱因斯坦用微分几何使“物质弯曲时空”的定性陈述精确，从而表明引力是这一基本思想的结果。相比之下，通过测量水星轨道的岁差，牛顿的引力平方反比定律被证明是错误的。因此，正确理解引力似乎需要理解曲率的性质，而曲率的性质目前是研究几何分析的数学家最直接追求的。黑洞，由广义相对论预测，现在已知存在，基本上是几何对象，并且一直是PI的大部分努力的焦点，从而产生了对这些迷人现象的更深入的物理见解的定理。鉴于几何分析在理解宇宙大尺度结构方面发挥着至关重要的作用，PI现在正在寻找几何动机来试图理解暗物质的性质。虽然已知暗物质占宇宙质量的23%，因此具有非常重要的引力效应，但它是不可见的。PI观察到的几何思想，以及其他动机，导致考虑实值标量场作为暗物质的模型，由爱因斯坦克莱因-戈登方程描述。天体物理学家已经观察到，暗物质的这个模型与星系的平坦旋转曲线是一致的。PI正在研究这样一种想法，即标量场暗物质中的密度波会在常规物质中产生密度波，从而导致星星的形成，以及一些星系中的条形和螺旋形图案，这是一种令人兴奋的可能性，得到了初步模拟的支持。如果是正确的，这将表明，虽然暗物质本身是不可见的，但它的引力效应可能相当引人注目。