Extremal metrics, the Calabi flow and related PDEs

该项目重点研究Kaehler几何中的极值度量、Calabi流和相关偏微分方程，如复杂Monge-Ampere方程和S. Donaldson最近提出的算子。Kaehler几何中的稳定性猜想将极值度量的存在性与底层复射影变的稳定性联系起来。稳定性猜想及其相关问题是Kaehler几何中的基本问题，它与偏微分方程、复分析和代数几何有着密切的联系。对卡拉比流的精确理解将有助于解决这样一个猜想。受Perelman在Hamilton问题中成功解决Ricci流的几何化猜想的启发，PI提出研究Calabi流来攻击极值度量的存在性。Calabi流的关键问题是其长时间存在性、渐近性及相关问题。在PI和他的合作者先前工作的基础上，他计划进一步研究卡拉比流在Kaehler表面上，特别是在Del Pezzo表面和环形表面上。PI还计划研究相关偏微分方程的正则性问题，例如复Monge-Ampere方程，该方程是Kaehler几何中的基本方程，与极值度量和Calabi流密切相关。特别是，PI计划研究在实Monge-Ampere方程中已经证明的某些正则性结果是否适用于复Monge-Ampere方程。对S. Donaldson算子的研究将会对这一问题有新的认识。Donaldson算子可以看作是一种同时具有实数和复蒙吉-安培算子性质的算子。由于我们试图从几何和物理的角度来理解几何偏微分方程，自然会产生一些问题，而几何偏微分方程与代数几何、复分析和数学物理等许多其他领域有着密切的关系。这些问题的关键特征之一是空间的整体结构如何影响此类方程解的局部解析性质。理解这一普遍原理将对物理学和其他科学领域产生广泛的影响。这项研究将对PI所在大学的学生产生立竿见影的效果。

This project focuses on the study of the extremal metrics in Kaehler geometry, the Calabi flow and related PDEs, such as the complex Monge-Ampere equation and an operator recently suggested by S. Donaldson. The stability conjecture in Kaehler geometry relates the existence of the extremal metrics to the stability of the underlying complex projective variety. The stability conjecture and related problems are fundamental problems in Kaehler geometry, which has tight relation with the partial differential equations, complex analysis and algebraic geometry. The precise understanding of the Calabi flow will lead to solving such a conjecture. Inspired by Perelman's success in Hamilton's problem to solve geometrization conjecture by Ricci flow, the PI proposes to study the Calabi flow to attack the existence of extremal metrics. The key issue for the Calabi flow is the long time existence, the asymptotic behavior and related problems. Based on the previous work of the PI and his collaborators, he plans to further study the Calabi flow on Kaehler surfaces, in particular on Del Pezzo surfaces and toric surfaces. The PI also plans to study the regularity problem of related PDEs, such as the complex Monge-Ampere equation, which is a fundamental equation in Kaehler geometry and is also closely related to the extremal metrics and the Calabi flow. In particular, the PI plans to study whether some regularity result, which has been proved for the real Monge-Ampere equation, holds or not for the complex Monge-Ampere equation. The study of S. Donaldson's operator will give new insight on this. Donaldson's operator can be viewed as an operator with some nature of both the real and complex Monge-Ampere operators. Problems in the proposal arise naturally from our attempts to understand geometric partial dierential equations from geometry and physics, which have tight relation with many other fields such as algebraic geometry, complex analysis and mathematical physics. One of the key features of these problems is how the global structure of a space influences the local, analytic properties of the solutions of such equations. Understanding this general principle will have broad impact in physics and other fields of science in general. The proposed research will have immediate beneficial effect on students in PI's home university.