Ricci flow of manifolds with singularities at infinity

无穷远奇点流形的 Ricci 流

• 批准号: EP/T019824/1
• 负责人: Peter Topping
• 金额: $ 46.21万
• 依托单位: University of Warwick
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2020
• 资助国家: 英国
• 起止时间: 2020 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FT019824%2F1
• 关键词: Ricci; flow; manifolds; singularities; infinity

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.

该提案涉及几何流，这是一个处于微分几何、分析、拓扑和非线性偏微分方程（PDE）理论的交叉学科。更具体地说，我们将考虑里奇流，这是一种采用弯曲空间（称为黎曼流形）并及时对其进行变形以使其更加均匀的方法。该场的重要性怎么强调都不为过。里奇流因解决一系列重大问题而闻名，例如悬赏 100 万美元的具有 100 年历史的庞加莱猜想和瑟斯顿几何猜想，但其应用的潜在范围远远不止于此。到目前为止，该理论几乎完全集中在紧致流形上，或者对其在无穷远处的行为有人为约束的流形，例如每个单位球体积的均匀上曲率界或正均匀下界。该提案针对下一波应用。为了实现这些，我们必须了解无穷远奇异的流动，为此，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。我们需要一系列的创新，包括新的曲率估计和对无限正曲流形几何的更好理解。即使在这些方面取得部分成功，也将改变该领域的适用性。进展将使我们了解开流形的几何和拓扑，而无需对其几何进行人为渐近约束。我们给出了一些主要开放问题的说明性例子，这些问题将属于我们设想的进步，例如丘的均匀化猜想，并描述实现这些问题的路线。该提案有一些雄心勃勃的目标。然而，它还包含一系列难度各异的猜想和问题，这些猜想和问题在许多方面都违背了理解无限曲率流动和无穷大坍缩行为的中心目标。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，使其变得可行。感谢几个国际团队的工作，包括 PI 和 M. Simon 在解决 3D 中的 Anderson-Cheeger-Colding-Tian 猜想时的工作，我们现在清楚地了解了所需的先验估计，该估计与迄今为止证明的尺度不变估计有很大不同，并且我们最终有了建立它们的路线图。

项目成果

Conformal tori with almost non-negative scalar curvature

• DOI: 10.1007/s00526-022-02220-9
• 发表时间: 2022-04
• 期刊: Calculus of Variations and Partial Differential Equations
• 影响因子: 2.1
• 作者: Jianchun Chu;Man-Chun Lee

$d_p$ convergence and $\epsilon$-regularity theorems for entropy and scalar curvature lower bounds.

• 发表时间: 2020-10
• 期刊: arXiv: Differential Geometry
• 作者: Man-Chun Lee;A. Naber;Robin Neumayer

K\"ahler manifolds and mixed curvature

• 发表时间: 2020-09
• 作者: Jianchun Chu;Man-Chun Lee;Luen-Fai Tam

Three-manifolds with non-negatively pinched Ricci curvature

具有非负收缩 Ricci 曲率的三流形

• DOI: 10.48550/arxiv.2204.00504
• 发表时间: 2022
• 作者: Lee M

Time Zero Regularity of Ricci Flow

• DOI: 10.1093/imrn/rnac300
• 发表时间: 2022-02
• 期刊: International Mathematics Research Notices
• 影响因子: 1
• 作者: Man-Chun Lee;P. Topping