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Collapsing in Differential Geometry and the Einstein Flow

微分几何的崩溃和爱因斯坦流

基本信息

批准号：
1810700
负责人：
John Lott
金额：
$ 24.21万
依托单位：
University of California-Berkeley
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-09-01 至 2022-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1810700&HistoricalAwards=false
关键词：
Collapsing Differential Geometry Einstein Flow

项目摘要

A geometric flow is a controlled way to smoothly deform a geometric object, such as a curve or a surface, or a 3-dimensional slice of spacetime equipped with a metric that measures length and angles. This project is about the Einstein flow, which deforms such slices in a way that, when the slices are stacked up, the result is 4-dimensional solution of the vacuum Einstein equations, the fundamental equations that govern our universe. Deducing something about the future (or past) state of the universe from information about the present is of evident interest in cosmology, and pose a very challenging mathematical problem. In this project, the investigator will focus on the asymptotic (or long-term) behavior of spacetimes, in the "cosmological" setting where the slices are compact (i.e., they wrap back on themselves, like the surface of a balloon, no matter how large their size). In this context, even if the spacetime does not develop singularities (e.g., a black hole), the spatial slices can asymptotically collapse, meaning that their volumes become smaller than a naive rescaling argument would suggest. Recently, the investigator adapted techniques from Riemannian geometry (and in particular the study of collapsing solutions of the Ricci flow) to give new information about expanding vacuum spacetimes. Part of this project will be to build on these results and obtain more precise results about the future asymptotics of these spacetimes. Another component of this project will be the training of graduate students and postdoctoral scholars. Results of this project will also be disseminated to the public via journal publications, conferences, and posting to online mathematics archives.Collapsing in differential geometry is the phenomenon that a sequence of Riemannian manifolds can converge to a lower dimensional space in the Gromov-Hausdorff topology. The research in this proposal will extend collapsing methods, both within differential geometry and within geometric flows. In recent work, the investigator adapted collapsing techniques to give new information about the future asymptotics of expanding Einstein flows. One feature was the avoidance of any a priori symmetry assumptions; instead, continuous symmetries appeared in the collapsing limit. The proposed research will extend this in various ways. One direction is a more concrete understanding of the asymptotics of expanding vacuum solutions. A second direction is the asymptotics of shrinking Einstein flows, which are relevant for spacetime singularities. In addition, the investigator will work on problems in differential geometry and geometric flows such as collapsing with a lower bound on the curvature operator, and the behavior of scalar curvature lower bounds under metric convergence.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

几何流是一种可控的方式，可以平滑地变形几何对象，例如曲线或曲面，或配备测量长度和角度的度量的三维时空切片。这个项目是关于爱因斯坦流的，它使这些切片变形，当切片堆叠起来时，结果是真空爱因斯坦方程的四维解，这是管理我们宇宙的基本方程。从现在的信息中推断出宇宙未来（或过去）的状态显然是宇宙学的兴趣所在，并提出了一个非常具有挑战性的数学问题。在这个项目中，研究人员将专注于时空的渐近（或长期）行为，在“宇宙学”的设置中，切片是紧凑的（即，它们会像气球的表面一样自己卷回去，不管它们的尺寸有多大）。在这种情况下，即使时空不发展奇点（例如，黑洞），空间切片可以渐近坍缩，这意味着它们的体积变得比天真的重新缩放论点所建议的要小。最近，研究人员采用了黎曼几何的技术（特别是里奇流坍缩解的研究）来提供有关扩展真空时空的新信息。该项目的一部分将是建立在这些结果的基础上，并获得关于这些时空未来渐近性的更精确的结果。该项目的另一个组成部分将是培训研究生和博士后学者。该项目的成果还将通过期刊出版物、会议和在线数学档案发布给公众。微分几何中的坍缩是一列黎曼流形在Gromov-Hausdorff拓扑中收敛到低维空间的现象。在这个建议中的研究将扩展崩溃的方法，无论是在微分几何和几何流。在最近的工作中，研究人员采用了塌缩技术，以提供有关扩展爱因斯坦流未来渐近性的新信息。一个特点是避免了任何先验对称性假设;相反，连续对称性出现在坍缩极限中。拟议的研究将以各种方式扩展这一点。一个方向是更具体地理解扩展真空解的渐近性。第二个方向是收缩爱因斯坦流的渐近性，这与时空奇点有关。此外，该研究员还将研究微分几何和几何流中的问题，例如曲率算子的下限崩溃，以及度量收敛下标量曲率下限的行为。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。