Extremal metrics, the Calabi flow and related PDEs

本项目主要研究Kaehler几何中的极值度量、Calabi流和相关的偏微分方程组，如复Monge-Ampere方程和S.Donaldson最近提出的一个算子。Kaehler几何中的稳定性猜想将极值度量的存在与基础复射影簇的稳定性联系起来。稳定性猜想及相关问题是Kaehler几何中的基本问题，它与偏微分方程组、复分析和代数几何有着密切的联系。对卡拉比流的准确理解将导致解决这样一个猜想。受Perelman在用Ricci流解决哈密尔顿问题解决几何猜想中的成功的启发，PI建议研究Calabi流以攻击极值度量的存在。Calabi流的关键问题是它的长时间存在性、渐近性态及相关问题。在PI及其合作者以前工作的基础上，他计划进一步研究Kaehler曲面上的Calabi流，特别是Del Pezzo曲面和Toric曲面上的Calabi流。PI还计划研究相关偏微分方程组的正则性问题，例如复Monge-Ampere方程，它是Kaehler几何中的一个基本方程，也与极值度量和Calabi流密切相关。特别是，PI计划研究一些已被证明适用于实Monge-Ampere方程的正则性结果是否适用于复Monge-Ampere方程。对S.Donaldson操作员的研究将为这一点提供新的见解。Donaldson算子可以被视为同时具有实算子和复数Monge-Ampere算子性质的算子。我们试图从几何和物理中理解几何偏微分方程组，而几何偏微分方程式与代数几何、复分析和数学物理等许多领域有着密切的联系，因此建议中的问题自然就产生了。这些问题的关键特征之一是空间的整体结构如何影响这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。这项拟议的研究将立即对Pi所在大学的学生产生有益影响。

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.