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

• 批准号: 2443749
• 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

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 被视为 EPSRC“几何与拓扑”研究领域最近取得突破的工具：它是庞加莱猜想和几何化猜想、可微球定理、Anderson-Cheeger 证明的基础。第三维的 Colding-Tian 猜想和广义 Smale 猜想 M. Simon 和 P. Topping 的证明第三维度中的 Anderson-Cheeger-Colding-Tian 猜想，Ricci Flow 用于平滑 Ricci 极限空间。这个方向的一个进一步开放的问题是 Ricci Flow 是否可以用于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实现了这一方向。Gianniotis-Schulze 的工作表明这是可能的。从具有孤立圆锥奇点的紧致流形开始构建 Ricci 流，该流形以正弯曲圆锥为模型。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 的结果扩展到以下情况。圆锥奇点沿着闭合曲线出现。该项目的成功完成将是开发正弯曲多面体空间流动方法的基础。