Collapsing in Differential Geometry and the Einstein Flow

几何流是一种可控制的方式，可以平滑地使几何对象变形，例如曲线或曲面，或者配备了测量长度和角度的度量衡的三维时空切片。这个项目是关于爱因斯坦流的，它以一种方式使这些薄片变形，当这些薄片堆叠起来时，结果是真空爱因斯坦方程的四维解，真空爱因斯坦方程是支配我们宇宙的基本方程。从有关现在的信息中推断出关于宇宙未来(或过去)状态的某些东西在宇宙学中显然是有兴趣的，并提出了一个非常具有挑战性的数学问题。在这个项目中，研究人员将重点放在时空的渐近(或长期)行为上，在“宇宙学”环境中，切片是致密的(即，它们像气球表面一样包裹着自己，无论它们有多大)。在这种情况下，即使时空没有发展出奇点(例如黑洞)，空间切片也可以渐近坍塌，这意味着它们的体积变得比天真的重新标度论点所暗示的要小。最近，这位研究人员采用了黎曼几何学的技术(特别是对Ricci流的塌缩解的研究)，给出了关于真空时空扩张的新信息。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准确的结果。该项目的另一个组成部分是研究生和博士后学者的培训。这个项目的结果也将通过期刊出版物、会议和在线数学档案发布给公众。微分几何中的折叠是指黎曼流形序列可以收敛到Gromov-Hausdorff拓扑中的低维空间的现象。这项提案中的研究将扩展折叠方法，既包括微分几何，也包括几何流动。在最近的工作中，研究人员采用了坍缩技术，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了任何先验的对称性假设；取而代之的是，在崩溃极限中出现了连续的对称性。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，研究人员将致力于微分几何和几何流动中的问题，如曲率算子下限的折叠，以及度量收敛下标量曲率下界的行为。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.