Complex Monge Ampere equation, the Kahler Einstein Problem and constant scalar metric problems

复蒙日安培方程、卡勒爱因斯坦问题和常标量度量问题

• 批准号: 1515795
• 负责人: Xiuxiong Chen
• 金额: $ 35.32万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-06-15 至 2019-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1515795&HistoricalAwards=false
• 关键词: Complex; Monge; Ampere; equation; Kahler

As a branch of Mathematics, Differential Geometry studies the shapes of spaces through distances and angles. The key concept involved is that of a so-called curvature, the simplest kind being the scalar curvature function on a space. The extremal cases are often the most interesting to study. In particular, the existence of a constant scalar curvature metric is of a major importance in Differential Geometry. It has a strong impact on other fields of sciences such as physics. For instance, the work of Calabi-Yau directly provided a mathematical foundation in mirror symmetry. According to A. Einstein, the theory of gravity can be interpreted as the geometry of the space-time. Therefore, the research in complex geometry is crucially important in physics and cosmology. The research proposed here also has impacts on algebraic geometry and partial differential equations. The project has an integrated education component which will enable the PI to continue supporting graduate students financially to pursue their research. In recent years, striking progress has been made in Kaehler geometry, particularly on the existence of the Kaehler-Einstein metrics and the limiting behavior of the Kaehler-Ricci flow solution, both in Fano manifolds. More exciting progress will follow after these works in this and adjacent area. In Kaehler geometry, the focus of the field is now on the existence of constant scalar curvature Kaehler metrics which is more general and harder than the existence of the Kaehler-Einstein metrics. This program on constant scalar curvature Kaehler metrics, which was first proposed by E. Calabi in 1950s, amounts to solving a 4th-order partial differential equation which naturally interacts with metric geometry as well as algebraic geometry. In this project, the PI will study a network of problems centering around the existence of constant scalar curvature metrics and other related areas. These include some fundamental problems in complex Monger-Ampere equations, a priori estimates for constant scalar curvature metrics, metric geometry as well as geometric flow (the Calabi flow and the Kaehler-Ricci flow).

作为数学的一个分支，微分几何通过距离和角度研究空间的形状。涉及的关键概念是所谓的曲率，最简单的一种是空间上的标量曲率函数。极端情况通常是最值得研究的。特别是，恒定标量曲率度量的存在在微分几何中非常重要。它对物理学等其他科学领域产生了重大影响。例如，卡拉比丘的工作直接提供了镜像对称的数学基础。根据爱因斯坦的说法，引力理论可以解释为时空的几何学。因此，复杂几何的研究在物理学和宇宙学中至关重要。这里提出的研究也对代数几何和偏微分方程产生影响。该项目有一个综合教育部分，这将使 PI 能够继续在经济上支持研究生进行研究。近年来，凯勒几何学取得了惊人的进展，特别是在法诺流形中凯勒-爱因斯坦度量的存在性和凯勒-里奇流解的极限行为方面。 在该领域及邻近领域开展这些工作之后，将会取得更令人兴奋的进展。在凯勒几何中，该领域现在的焦点是恒定标量曲率凯勒度量的存在，它比凯勒-爱因斯坦度量的存在更普遍、更困难。这个关于恒定标量曲率凯勒度量的程序由 E. Calabi 在 1950 年代首次提出，相当于求解一个与度量几何和代数几何自然相互作用的四阶偏微分方程。在这个项目中，PI 将研究围绕常标量曲率度量和其他相关领域的存在的问题网络。其中包括复杂 Monger-Ampere 方程中的一些基本问题、常标量曲率度量的先验估计、度量几何以及几何流（卡拉比流和 Kaehler-Ricci 流）。