Ricci flow of manifolds with singularities at infinity

这项建议涉及几何流动，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)理论之间的学科。更具体地说，我们将考虑Ricci流，这是一种将空间弯曲，称为黎曼流形，并及时将其变形以使其更均匀的方法。场的重要性怎么强调都不为过。Ricci Flow以解决一系列重大问题而闻名，比如有100年历史的庞卡莱猜想(该猜想有100万美元的悬赏)和瑟斯顿的几何猜想，但它的潜在应用范围远远不止于此。到目前为止，该理论几乎只关注紧致的流形，或者对其在无穷远处的行为有人为限制的流形，例如每个单位球的体积的一致曲率上界或正一致下界。这项提议是针对下一波应用的。要实现这些，我们必须了解在无穷远处奇异的流动，而要做到这一点，我们将需要推进非线性偏微分方程组的理论，并更好地了解它们与几何的相互作用。我们将需要一系列的创新，包括新的曲率估计和对正曲线流形无穷远处的几何更好的理解。即使沿着这些线取得的部分成功也将改变该领域的适用性。进展将使我们了解开流形的几何和拓扑，而不需要对其几何进行人为的渐近约束。我们列举了一些主要的公开问题的说明性例子，这些问题将落到我们所预见的进展中，例如邱氏的统一化猜想，并描述了实现这些问题的途径。然而，它也包含了一系列猜想和问题，难度各不相同，这些猜想和问题在许多方面都与理解无限曲率的流动和无穷远处的崩溃行为的核心目标背道而驰。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使这一方向可行的基础理论。由于几个国际团队的工作，包括PI和M.Simon在他们解决Anderson-Cheeger-Colding-Tian猜想的3D中的工作，我们现在对所需的先验估计有了明确的想法，这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.