Collapsing in Differential Geometry and the Einstein Flow

几何流是一种受控的方式，可以平滑地变形一个几何物体，比如曲线或表面，或者三维时空切片，配备了测量长度和角度的度量。这个项目是关于爱因斯坦流的，它以某种方式使这样的切片变形，当这些切片堆叠起来时，结果是真空爱因斯坦方程的四维解，即控制我们宇宙的基本方程。从现在的信息中推断出宇宙的未来（或过去）状态显然是宇宙学的兴趣所在，也是一个非常具有挑战性的数学问题。在这个项目中，研究者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，它们像气球表面一样包裹在自己身上，不管它们有多大）。在这种情况下，即使时空没有发展出奇点（例如，黑洞），空间片也可以渐近坍缩，这意味着它们的体积比天真的重新缩放论证所建议的要小。最近，研究者利用黎曼几何的技术（特别是对里奇流坍缩解的研究）给出了关于膨胀真空时空的新信息。这个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养研究生和博士后学者。该项目的结果也将通过期刊出版物、会议和在线数学档案向公众发布。在微分几何中，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。本提案中的研究将扩展在微分几何和几何流中的坍缩方法。在最近的工作中，研究人员对坍缩技术进行了改进，以提供有关爱因斯坦流扩展的未来渐近性的新信息。其中一个特点是避免任何先验的对称假设；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膨胀真空解的渐近性。第二个方向是缩小的爱因斯坦流的渐近性，这与时空奇点有关。此外，研究者将研究微分几何和几何流中的问题，如曲率算子的下界坍缩，以及度量收敛下标量曲率下界的行为。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.