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Ricci flow from spaces with edge type conical singularities

来自具有边缘型圆锥奇点的空间的利玛窦流

基本信息

批准号：
2443749
负责人：
金额：
--
依托单位：
University of Warwick
依托单位国家：
英国
项目类别：
Studentship
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=studentship-2443749
关键词：
Ricci flow spaces edge type

项目摘要

Ricci Flow has seen been a tool in recent breakthroughs in the EPSRC research area of 'Geometry and Topology': It was fundamental in the proof of the Poincaré and Geometrization Conjectures, the Differentiable Sphere Theorem, the proof of the Anderson-Cheeger-Colding-Tian conjecture in dimension three, and the Generalized Smale Conjecture. In work of M. Simon and P. Topping on the proof of the Anderson-Cheeger-Colding-Tian conjecture in dimension three, Ricci Flow was used to smooth out Ricci limit spaces. A further open question in this direction is if Ricci Flow can be used to smooth out positively curved polyhedral spaces in higher dimension. First steps in this direction have recently been achieved by Bamler-Cabezas-Rivas-Wilking and Gianniotis-Schulze with different approaches. The work of Gianniotis-Schulze shows that it possible to construct a Ricci Flow starting from a compact manifold with isolated conical singularities which are modelled on positively curved cones. As a first step towards flowing from polyhedral spaces, Lucas Lavoyer de Miranda is working on extending the results of Gianniotis-Schulze to the case that the conical singularities occur along a closed curve. There are several new ideas and techniques to be developed. Successful completion of this project will be fundamental in developing an approach to flowing from positively curved polyhedral spaces.

Ricci Flow 已成为 EPSRC“几何与拓扑”研究领域最近取得突破的工具：它是证明庞加莱猜想和几何化猜想、可微球定理、证明第三维 Anderson-Cheeger-Colding-Tian 猜想以及广义 Smale 猜想的基础。在 M. Simon 和 P. Topping 证明第三维 Anderson-Cheeger-Colding-Tian 猜想的工作中，Ricci Flow 被用来平滑 Ricci 极限空间。在这个方向上另一个悬而未决的问题是，里奇流是否可以用来平滑更高维度的正弯曲多面体空间。 Bamler-Cabezas-Rivas-Wilking 和 Gianniotis-Schulze 最近通过不同的方法朝着这个方向迈出了第一步。 Gianniotis-Schulze 的工作表明，可以从具有孤立圆锥奇点（以正曲锥体为模型）的紧凑流形开始构建 Ricci 流。作为从多面体空间流动的第一步，Lucas Lavoyer de Miranda 正在努力将 Gianniotis-Schulze 的结果扩展到圆锥奇点沿着闭合曲线出现的情况。有一些新的想法和技术有待开发。该项目的成功完成对于开发从正弯曲多面体空间流动的方法至关重要。