Ricci flow from spaces with edge type conical singularities

Ricci流是EPSRC研究领域“几何与拓扑学”最近取得突破的一个工具：它是证明Poincaré和几何化猜想、可微球定理、三维Anderson-Cheeger-Colding-Tian猜想的证明以及推广的斯梅尔猜想的基础。在M.Simon和P.Topping在证明三维Anderson-Cheeger-Colding-Tian猜想的工作中，利用Ricci流来光滑Ricci极限空间。在这个方向上的另一个悬而未决的问题是，Ricci流是否可以用于在更高维度上平滑正曲线多面体空间。Bamler-Cabezas-Rivas-Wilking和Gianniotis-Schulze最近以不同的方法朝着这一方向迈出了第一步。Gianniotis-Schulze的工作表明，从具有孤立锥形奇点的紧致流形开始构造Ricci流是可能的，这些锥形奇点是以正曲线锥体为模型的。作为从多面体空间流动的第一步，卢卡斯·拉沃耶·德·米兰达正致力于将Gianniotis-Schulze的结果推广到圆锥奇点沿闭合曲线发生的情况。有几个新的想法和技术有待开发。这个项目的成功完成将是开发一种从正曲线多面体空间流动的方法的基础。

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.