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

首先，PI将继续他对标量曲率的研究，特别是对3流形的研究。PI在这一领域的先前成果包括2002年与Andre Neves的联合工作，该工作对Yamabe不变量大于RP^3的素数3流形进行了分类，以及2008年与Pengzi Miao的一篇论文，该论文给出了具有非负标量曲率的3流形中曲面容量的上界。2009年，PI与Simon Brendle， Michael eichmaair和AndreNeves联合发表论文，证明了包含内嵌不可压缩RP^2的紧3流形上的{A＿minR＿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。利用里奇流，证明了3流形在相等的情况下是球面空间形式。其次，PI将继续努力证明完整的彭罗斯猜想。PI在2001年的论文中证明了三维的黎曼彭罗斯猜想，将惠斯肯和伊尔曼证明的一个黑洞的情况改进为使用不同的技术证明任意数量的黑洞。此后，PI在2005年证明了零面积奇点的类似不等式（带有一些额外的假设），2007年在与Dan Lee的联合工作中证明了小于8维的黎曼彭罗斯猜想，并在2009年与Marcus Khuri的联合工作中证明了柯西数据（M^3,g,k）上的完整彭罗斯猜想简化为黎曼情况，只要某些p.d.e.s系统可以求解。这些p.d.e.系统依赖于一个新的恒等式，他们证明了这个恒等式叫做广义Schoen-Yau恒等式，他们相信这个恒等式对于数学相对论中的许多问题都是非常有用的。第三，PI正在为他自己开辟一个新的研究方向，因为他检查广义相对论的公理，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，几十年前他们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。爱因斯坦使用微分几何使“物质弯曲时空”这一定性表述变得精确，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进动，牛顿的引力平方反比定律被证明是错误的。因此，正确地理解引力似乎需要理解曲率的性质，这是目前研究几何分析的数学家最直接追求的。广义相对论预言了黑洞的存在，现在已知黑洞的存在，基本上是几何物体，一直是PI努力的重点，产生了对这些迷人现象有更深物理见解的定理。鉴于几何分析的丰富历史在理解宇宙的大尺度结构中起着至关重要的作用，PI现在正在寻找几何动机，试图理解暗物质的本质。而暗物质则占了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.

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.