Geometric Flows and Canonical Kahler Metrics

奖项：DMS 1710500，首席研究员：郭斌来自引力理论的爱因斯坦方程有一个几何解释，选择一个首选的度量来确定空间上的长度和角度。 这些方程的可解性条件的确定最近取得了进展，并继续在几何中提出重要的开放问题。将要进行的项目包括通过几何流来解决这些问题的方法，这些几何流使空间上的度量在可能导致规范度量的方向上变形，或者，可能会发展出一个奇点，阻碍这种度量的进展。其中一些项目将研究凯勒的长时间和短时间奇点的形成-Ricci流与Song和Tian提出的解析极小模型程序密切相关。锥形Kaehler-Einstein方程在Fano流形上已经非常成功，这些项目将研究代数簇中锥形正则Kaehler度量的存在性和性质，并探索它们在解决代数几何中的公开问题中的应用。 另一项工作将研究抛物型耦合系统，它起源于物理学中的广义相对论。耦合系统的解将产生爱因斯坦真空度规，并揭示底层空间的结构。

Award: DMS 1710500, Principal Investigator: Bin GuoThe Einstein equations from gravitational theory have a geometric interpretation that picks out a preferred metric to determine lengths and angles on a space. The determination of conditions for solvability of those equations has seen recent progress and continues to pose important open problems in geometry. The projects to be carried out include approaches to those problems through geometric flows that deform the metric on a space in a direction that might lead to a canonical metric, or, alternatively, might develop a singularity that blocks progress toward such a metric.Some of these projects will study the formation of long-time and short-time singularities from Kaehler-Ricci flow which are closely related to the analytic minimal model program proposed by Song and Tian. The conical Kaehler-Einstein equations have been very successful for Fano manifolds, and these projects will investigate the existence and properties of conical canonical Kaehler metrics in algebraic varieties and explore their applications in solving open problems in algebraic geometry. Another line of work will study coupled systems of parabolic type which originated from general relativity in physics. The solutions to the coupled systems will yield Einstein vacuum metrics and shed light on the structure of the underlying space.