Geometric Flows and Canonical Kahler Metrics

几何流和规范卡勒度量

• 批准号: 1945869
• 负责人: Bin Guo
• 金额: $ 8.79万
• 依托单位: Rutgers University Newark
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-07-01 至 2020-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1945869&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.

奖项：DMS 1710500，首席研究员：郭斌引力理论中的爱因斯坦方程具有几何解释，可以选择一个首选度量来确定空间上的长度和角度。 这些方程可解条件的确定最近取得了进展，并继续在几何学中提出重要的开放问题。即将开展的项目包括通过几何流解决这些问题的方法，这些几何流使空间上的度量变形，从而可能导致规范度量，或者可能形成阻碍此类度量进展的奇点。这些项目将研究Kaehler-Ricci流的长时和短时奇点的形成，这与宋和田提出的解析最小模型程序密切相关。圆锥凯勒-爱因斯坦方程对于法诺流形来说非常成功，这些项目将研究代数簇中圆锥正则凯勒度量的存在性和性质，并探索它们在解决代数几何中的开放问题中的应用。 另一项工作将研究源自物理学中广义相对论的抛物线型耦合系统。耦合系统的解决方案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

On the degeneration of asymptotically conical Calabi–Yau metrics

渐近圆锥形卡拉比丘度规的退化

• DOI: 10.1007/s00208-021-02229-z
• 发表时间: 2021-07
• 期刊: Mathematische Annalen
• 影响因子: 1.4
• 作者: Collins; T.
• 通讯作者: T.