Canonical metrics in complex geometry

该奖项是根据2009年《美国复苏和再投资法案》(公法111-5)提供资金的。PI提出的研究是关于代数簇上的规范Kahler度量的研究。在过去的十年里，人们对极值度量的存在与基本代数簇的稳定性之间的关系进行了深入的猜测，并对这种关系进行了深入的研究。PI研究的一个主要方面是，当不存在极值度量时，人们可以说什么。这个问题将在代数方面和度量方面进行研究，其中包括将许多现有的关于极值度量的结果推广到沿因子具有尖点奇点的非紧流形。PI建议的另一个方向是使用几何流来攻击正则度量的存在。这里最吸引人的方面之一是研究当不存在极值度量时可以形成什么样的奇点，PI将建立在他早期关于直纹曲面和环面簇上的Calabi流以及Kahler-Ricci流的工作的基础上。几何偏微分方程组支配着物理世界的大部分。例如，爱因斯坦方程的解与我们对宇宙的理解密切相关。这项拟议的研究研究与爱因斯坦方程有关的微分方程，关键问题是空间的整体结构如何影响这类方程解的局部解析性质。了解这一现象将在物理学和一般科学中得到应用。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The PI's proposed research is concerned with the study of canonical Kahler metrics on algebraic varieties. There are deep conjectures relating the existence of extremal metrics to stability of the underlying algebraic variety and understanding this relationship has been studied intensively in the last decade. One major aspect of the PI's research is what one can say when no extremal metric exists. This problem will be studied both on the algebraic side to understand how varieties can be destabilized, and also in terms of metrics, which involves extending many of the existing results on extremal metrics to non-compact manifolds with cusp-like singularities along divisors.Another direction in the PI's proposal is the use of geometric flows to attack the existence of canonical metrics. Here too one of the most fascinating aspects is to study what kinds of singularities can form when no extremal metric exists and the PI will build on his earlier work on the Calabi flow on ruled surfaces and toric varieties and on the Kahler-Ricci flow.Geometric partial differential equations govern much of the physical world. For example solutions of Einstein's equations are intimately related to our understanding of the universe. The proposed research studies differential equations related to Einstein's equations and the key question is how the global structure of a space influences the local, analytic properties of the solutions of such equations.Understanding this phenomenon will have applications in physics and the sciences in general.