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

微分几何是数学的一个分支，它通过距离和角度来研究空间的形状。所涉及的关键概念是所谓的曲率，最简单的是空间上的标量曲率函数。极端情况往往是最有趣的研究。特别是，存在一个常数标量曲率度量是一个重大的重要性，在微分几何。它对物理学等其他科学领域产生了巨大的影响。例如，卡-丘的工作直接提供了镜像对称的数学基础。根据A.爱因斯坦的引力理论可以解释为时空的几何学。因此，复几何的研究在物理学和宇宙学中具有极其重要的意义。这里提出的研究也有影响代数几何和偏微分方程。该项目有一个综合的教育部分，这将使PI继续资助研究生进行研究。近年来，Kaehler几何的研究取得了令人瞩目的进展，特别是在Fano流形中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，将有更多令人兴奋的进展。在Kaehler几何中，该领域的焦点现在是常数标量曲率Kaehler度量的存在性，这比Kaehler-Einstein度量的存在性更普遍和更困难。这个程序是关于常数量曲率Kaehler度量的，它是由E. Calabi在20世纪50年代提出了一个四阶偏微分方程，它与度量几何和代数几何自然地相互作用。在这个项目中，PI将研究围绕常数标量曲率度量和其他相关领域存在的问题网络。这些问题包括复Monger-Ampere方程的一些基本问题、常数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).