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，差异几何形状通过距离和角度研究了空间的形状。涉及的关键概念是所谓的曲率，最简单的是空间上标量曲率函数。极端情况通常是最有趣的研究。特别是，恒定标量曲率度量的存在在差异几何形状中至关重要。它对其他科学领域（例如物理学）有很大的影响。例如，卡拉比尤（Calabi-Yau）的工作直接在镜面对称性中提供了数学基础。根据爱因斯坦A.的说法，重力理论可以解释为时空的几何形状。因此，复杂几何形状的研究在物理和宇宙学上至关重要。这里提出的研究还对代数几何和部分微分方程产生了影响。该项目具有综合的教育组成部分，这将使PI能够继续在经济上为研究生提供支持以进行研究。近年来，在Kaehler的几何形状中取得了惊人的进步，特别是在Kaehler-Einstein指标的存在以及Kaehler-Irci流动溶液的限制行为上，均在FANO歧管中。 在这些工作和邻近地区的这些作品之后，将随着更令人兴奋的进步。在Kaehler的几何形状中，该领域的重点是存在恒定标态曲率Kaehler指标的存在，这比Kaehler-Einstein指标的存在更一般和更难。该程序关于恒定标态曲率Kaehler指标，该计划是由E. calabi在1950年代首次提出的，等于求解一个四阶偏微分方程，该方程自然与公式的几何相互作用以及代数的几何形状。在该项目中，PI将研究围绕恒定标量曲率指标和其他相关领域的存在的问题网络。这些包括在复杂的贩运方程中的一些基本问题，对恒定标量曲率指标，度量几何形状以及几何流量（卡拉比流量和kaehler-ricci流）的先验估计值。