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）资助的。PI提出的研究是关于代数簇上的规范Kahler度量的研究。有深刻的abstractures相关的极值度量的存在下的代数簇的稳定性和理解这种关系已被深入研究，在过去的十年。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进行研究，以了解如何品种可以不稳定，也在度量方面，这涉及到扩展许多现有的结果极值度量的非紧流形与尖状奇点沿着因子。另一个方向在PI的建议是使用几何流攻击的存在性规范度量。这里也是最迷人的方面之一是研究什么样的奇点可以形成时，没有极值度量存在和PI将建立在他的早期工作卡拉比流的直纹曲面和复曲面品种和卡勒-里奇流。几何偏微分方程治理大部分的物理世界。例如，爱因斯坦方程的解与我们对宇宙的理解密切相关。 该研究主要研究与爱因斯坦方程相关的微分方程，其关键问题是空间的整体结构如何影响这些方程解的局部解析性质。理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.