Ricci flow of manifolds with singularities at infinity

这个建议涉及几何流，这是一个位于微分几何，分析，拓扑学和非线性偏微分方程（PDE）理论的接口的主题。更具体地说，我们将考虑里奇流，这是一种将弯曲空间（称为黎曼流形）在时间上变形以使其更均匀的方法。场的重要性怎么强调都不过分。里奇流是著名的解决一系列重大问题，如100岁的庞加莱猜想，其中有$1，000，000的赏金附加到它，和瑟斯顿的几何化猜想，但其应用的潜在范围远远超出。到目前为止，该理论几乎完全集中在紧致的流形上，或者在无穷远处对其行为有人为约束的流形上，例如每个单位球的体积上的一致曲率上界或正一致下界。这一提议是针对下一波应用的。为了实现这些，我们必须理解在无穷远处奇异的流动，要做到这一点，我们需要推进非线性偏微分方程的理论，并更好地理解它们与几何的相互作用。我们需要一系列创新，包括新的曲率估计和对正弯曲流形无穷远处几何的更好理解，即使沿着这些方面取得部分成功，也将改变该领域的适用性。进展将使我们了解的几何和拓扑的开放流形没有人为的渐近约束其几何。我们给出了一些说明性的例子，主要的开放问题，将下降到我们设想的进展，如丘的均匀化猜想，并描述了一条路线来实现它们。然而，它也包含了一系列不同难度的理论和问题，这些理论和问题在许多方面都与理解无限大曲率流动和坍缩行为的中心目标背道而驰。这个研究方向特别令人兴奋的是，只有在过去的几年里，我们才成功地发展了基础理论，使其可行。感谢多个国际团队的工作，包括PI和M。Simon在他们对3D中Anderson-Cheeger-Colding-Tian猜想的解决方案中，我们现在对所需的先验估计有了一个清晰的概念，它与迄今为止证明的尺度不变估计有很大的不同，我们终于有了建立它们的路线图。

This proposal concerns geometric flows, which is a subject that lies at the interface of differential geometry, analysis, topology and the theory of nonlinear partial differential equations (PDEs). More specifically, we will consider Ricci flow, which is a way of taking a curved space, known as a Riemannian manifold, and deforming it in time to make it more uniform.The importance of the field cannot be overstated. Ricci flow is famous for solving a string of major problems such as the 100 year old Poincaré conjecture, which had a $1,000,000 bounty attached to it, and Thurston's geometrisation conjecture, but the potential extent of its applications lies far beyond. Up until now, the theory has focussed almost exclusively on manifolds that are compact, or that have artificial constraints on their behaviour at infinity such as a uniform upper curvature bound or a positive uniform lower bound on the volume of every unit ball. This proposal is directed towards the next wave of applications. To realise these we must understand flows that are singular at infinity, and to do this we will need to advance the theory of nonlinear PDEs and understand better their interaction with geometry. We will require a collection of innovations, including new curvature estimates and a better understanding of the geometry at infinity of positively curved manifolds.Even partial success along these lines will transform the applicability of the field. Progress will give us an understanding of the geometry and topology of open manifolds without artificial asymptotic constraints on their geometry. We give some illustrative examples of major open problems that would fall to the advances that we envisage, such as Yau's Uniformisation Conjecture, and describe a route to achieve them.The proposal has some highly ambitious objectives. However, it also contains a collection of conjectures and problems, of varying difficulty, that push on many fronts against the central aim of understanding flows with unbounded curvature, and collapsing behaviour, at infinity. What is particularly exciting about this research direction is that only in the past few years have we been successful in developing the foundational theory to make this feasible. Thanks to the work of several international teams, including that of the PI and M. Simon in their resolution of the Anderson-Cheeger-Colding-Tian conjecture in 3D, we now have a clear idea of the required a priori estimates, which differ substantially from the scale-invariant estimates proved thus far, and we finally have a roadmap towards establishing them.