Canonical metrics in Kahler geometry

卡勒几何中的规范度量

• 批准号: 0905873
• 负责人: Jacob Sturm
• 金额: $ 38.83万
• 依托单位: Rutgers University Newark
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-01 至 2016-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0905873&HistoricalAwards=false
• 关键词: Canonical; metrics; Kahler; geometry

The PI plans to continue research in three inter-related areas. The first concerns geometric approximation of Kaehler metrics by Bergman metrics. The work of Tian, Yau and Zelditch shows that Kaehler metrics can be smoothly approximated by Bergman metrics, and this ``quantization processhas found numerous striking applications to Kaehler geometry. The PI, in joint work with Phong, has shown that this approximation holds as well at the geometric level: thus, geodesic segments and geodesic rays can be canonically approximated by segments and rays in the space of Bergman metrics. The first project is aimed at deepening our understanding of the structure of the space of Kaehler metrics: The PI will investigate higher dimensional analogues of such Bergman approximations using methods of PDE and pluri-potential theory. The second project concerns K-stability and constant scalar curvature metrics. Here the motivating question is a conjecture of Donaldson which asserts that the non-existence of a csc metric should imply the existence of a geodesic ray along which the K-energy decreases. The PI will study this via the geodesic rays associated to test configurations (constructed in joint work with Phong). The next key step in this program is to prove C^{1,1} regularity of these rays using some of the recent advances in pluri-potential theory. The third project concerns the Kaehler-Ricci flow. The goal is to relate the S-condition and the B-condition (introduced in joint work with Phong, and studied as well in joint work with Song and Weinkove) to some of the more classical notions of stability in algebraic geometry.The main theme of this proposal concerns the Einstein equation, which originally arose in the theory of general relativity. The great insight from physics is that many of the mysteries of the universe can be explained if one accepts the notion that our universe is curved. This conceptual breakthrough may be compared to the discovery, dating back to the early Greek philosophers, that the surface of the earth is curved. The Einstein equation gives a mathematical formulation of the curvature properties of the universe which is precise enough to make accurate predictions for a vast range of large scale physical phenomena. It turns out, for reasons that remain quite obscure, that the same Einstein equations can also be used to resolve deep and longstanding problems in topology and geometry. The partial differential equations that arise in this approach are non-linear, and their study requires a broad range of tools from real and complex analysis, as well as algebraic and differential geometry. This project will investigate the interplay between these various branches of mathematics, with the goal of furthering the understanding of fundamental geometric structures.

PI计划在三个相互关联的地区继续研究。第一个涉及伯格曼指标对Kaehler指标的几何近似。田，Yau和Zelditch的工作表明，伯格曼指标可以平稳地近似Kaehler指标，并且此``kealitization``量化processhas都发现了对Kaehler几何形状的许多惊人应用。 PI在与Phong的联合工作中表明，这种近似值在几何水平上也保持不变：因此，在伯格曼指标空间中的片段和射线可以在典型上近似地球段和地球射线。第一个项目的目的是加深我们对Kaehler指标空间结构的理解：PI将使用PDE和PLURI-POTICAL理论的方法研究此类Bergman近似值的更高维度类似物。第二个项目涉及K稳定性和恒定标量曲率指标。在这里，激励的问题是唐纳森的猜想，它断言CSC指标的不存在应暗示存在k-Energy降低的地理射线。 PI将通过与测试构型相关的测量光线（与Phong的联合工作构建）一起研究。该程序的下一个关键步骤是使用Pluri-Potential理论的最新进展证明这些射线的c^{1,1}。第三个项目涉及Kaehler-Ricci流。目的是将S条件和B条件（在与Phong的联合合作中引入，并在与Song and Weinkove的联合合作中进行研究）与代数几何的一些更古典的稳定性概念进行了研究。该建议的主要主题涉及Einstein方程，而Einstein方程最初是在一般相互关系理论中的原始主题。物理学的深刻见解是，如果人们接受我们宇宙弯曲的观念，可以解释宇宙的许多奥秘。可以将这种概念的突破与发现可以追溯到早期的希腊哲学家的发现，即地球的表面是弯曲的。爱因斯坦方程给出了宇宙曲率特性的数学表述，这足够精确，可以对广泛的大规模物理现象进行准确的预测。事实证明，出于相当晦涩的原因，同一爱因斯坦方程也可以用于解决拓扑和几何形状方面的深层且长期存在的问题。在这种方法中出现的部分微分方程是非线性的，他们的研究需要来自真实和复杂分析以及代数和差异几何形状的广泛工具。该项目将研究数学的各个分支之间的相互作用，以进一步了解基本几何结构。