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

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。所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。极端的案例往往是最值得研究的。特别地，常数量曲率度量的存在在微分几何中具有重要意义。它对物理学等其他科学领域产生了强烈的影响。例如，Calabi-Yau的工作直接提供了镜像对称的数学基础。根据爱因斯坦的观点，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都是至关重要的。本文提出的研究对代数几何和偏微分方程组也有影响。该项目有一个综合教育部分，这将使PI能够继续为研究生继续进行研究提供财政支持。近年来，Kaehler几何取得了显著的进展，特别是在Fano流形上的Kaehler-Einstein度规的存在性和Kaehler-Ricci流解的极限行为方面。在这一地区和邻近地区的这些工作之后，将会有更令人兴奋的进展。在Kaehler几何中，目前该领域的焦点是常标量曲率Kaehler度量的存在，它比Kaehler-Einstein度量的存在更普遍，也更困难。这个由E.Calabi在20世纪50年代首次提出的关于常标量曲率Kaehler度量的程序，相当于求解一个自然地与度量几何和代数几何相互作用的四阶偏微分方程。在这个项目中，PI将围绕常标量曲率度量的存在性和其他相关领域来研究网络问题。这些问题包括复蒙格-安培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(Calabi流和Kaehler-Ricci流)。

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).