Geometric Flows and Canonical Kahler Metrics

几何流和规范卡勒度量

• 批准号: 1710500
• 负责人: Bin Guo
• 金额: $ 14.77万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-08-01 至 2019-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1710500&HistoricalAwards=false
• 关键词: Geometric; Flows; Canonical; Kahler; Metrics

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.

获奖：DMS 1710500，首席研究员：国斌。爱因斯坦引力方程有一个几何解释，它选择了一个首选度量来确定空间的长度和角度。这些方程可解条件的确定最近取得了进展，并继续构成几何中的重要未决问题。要实施的项目包括通过几何流来解决这些问题的方法，这些几何流会使空间上的度量在一个方向上变形，从而可能导致规范度量，或者，或者，可能会发展出一个奇点，从而阻碍向这种度量的进展。其中一些项目将研究Kaehler-Ricci流的长时间和短时间奇点的形成，这与Song和Tian提出的解析最小模型程序密切相关。圆锥Kaehler- einstein方程对于Fano流形已经非常成功，这些项目将研究圆锥正则Kaehler度量在代数变量中的存在性和性质，并探索其在求解代数几何开放问题中的应用。另一项工作是研究物理学中起源于广义相对论的抛物型耦合系统。耦合系统的解将产生爱因斯坦真空度量，并揭示底层空间的结构。

项目成果

A Geometric Construction of Solutions to 11D Supergravity

11D 超重力解的几何构造

• 发表时间: 2019
• 期刊: Communications in Mathematical Physics
• 影响因子: 2.4
• 作者: Fei, Teng;Guo, Bin;Phong, Duong H.
• 通讯作者: Phong, Duong H.

On convergence criteria for the coupled flow of Li–Yuan–Zhang

• DOI: 10.1007/s00209-019-02272-2
• 发表时间: 2018-08
• 期刊: Mathematische Zeitschrift
• 影响因子: 0.8
• 作者: Teng Fei;B. Guo;D. Phong

Pseudo-locality for a coupled Ricci flow

• DOI: 10.4310/cag.2018.v26.n3.a5
• 发表时间: 2015-10
• 期刊: arXiv: Differential Geometry
• 作者: B. Guo;Zhijie Huang;D. Phong