Elliptic and parabolic complex Monge-Ampere equations on compact manifolds

紧流形上的椭圆和抛物线复数 Monge-Ampere 方程

• 批准号: 1332196
• 负责人: Benjamin Weinkove
• 金额: $ 6.14万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-11-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1332196&HistoricalAwards=false
• 关键词: Elliptic; parabolic; complex; Monge; Ampere

The PI plans to investigate projects in three areas, related by the theme of the complex Monge-Ampere equation. The first project will build on the PI's work with Tosatti which gave an analogue of Yau's Theorem for compact Hermitian manifolds. The PI proposes to generalize these estimates and apply them to obtain new results on the Bott-Chern space of a complex manifold. The second project deals with the Kahler-Ricci flow - which can be regarded as the parabolic complex Monge-Ampere equation - on algebraic varieties. As part of a program to understand the analytic minimal model program, the PI and Song investigated the Kahler-Ricci flow on algebraic varieties and gave necessary conditions under which the flow contracts an exceptional divisor. The PI proposes to extend these results to deal with more general singularities and also to study the behavior of the curvature tensor along the Kahler-Ricci flow. The final project concerns the Calabi-Yau equation on symplectic manifolds. Donaldson conjectured that the complex Monge-Ampere equation solved by Yau has an analogue for symplectic 4-manifolds with compatible almost complex structures.He gave applications of his conjecture to symplectic topology. Special cases of Donaldson's conjecture were established by the PI and his co-authors. The PI will undertake an analysis of the structure of the blow-up set for the Calabi-Yau equation, with the ultimate goal of proving Donaldson's conjecture.An important problem in mathematics, and physics, is to understand the interaction between geometry and differential equations. This proposal concerns a well-known and centuries-old mathematical object called the Monge-Ampere equation. This equation arises naturally in the study of geometry and is closely related to Einstein's equations in physics. A main goal of this project is to find applications of the Monge-Ampere equation to more general and commonly occuring geometric objects where it was not previously known that a connection exists. By finding new relationships between this classical differential equation and geometry, the PI aims to further our understanding of what kind of geometric structures can exist. In addition, the Monge-Ampere equation is deeply related to geometry via an associated heat flow. It is expected that this heat flow will help us understand some old and difficult problems concerning solutions to algebraic equations. Indeed, the solutions of algebraic equations define geometric objects, and the heat flow deforms these objects and can extract information from them. The PI will investigate the precise behavior of the Monge-Ampere heat flow on such geometric objects.

PI 计划调查三个领域的项目，这些领域与复杂的 Monge-Ampere 方程的主题相关。 第一个项目将建立在 PI 与 Tosatti 的合作基础上，该工作给出了紧致埃尔米特流形的丘氏定理的模拟。 PI 建议推广这些估计并应用它们来获得复杂流形的 Bott-Chern 空间上的新结果。 第二个项目涉及代数簇上的卡勒-里奇流（Kahler-Ricci flow）——它可以被视为抛物线复数 Monge-Ampere 方程。 作为理解解析最小模型程序的一部分，PI 和 Song 研究了代数簇上的 Kahler-Ricci 流，并给出了该流收缩异常除数的必要条件。 PI 建议扩展这些结果以处理更一般的奇点，并研究曲率张量沿 Kahler-Ricci 流的行为。 最终项目涉及辛流形上的 Calabi-Yau 方程。 唐纳森猜想丘所解的复Monge-Ampere方程可以类比具有兼容的近复结构的辛4流形。他把他的猜想应用到了辛拓扑中。唐纳森猜想的特例是由 PI 和他的合著者建立的。 PI将对Calabi-Yau方程的爆炸集结构进行分析，最终目标是证明唐纳森猜想。数学和物理中的一个重要问题是理解几何和微分方程之间的相互作用。 该提议涉及一个众所周知的、有数百年历史的数学对象，称为蒙日-安培方程。 这个方程在几何研究中自然出现，与物理学中的爱因斯坦方程密切相关。 该项目的主要目标是找到 Monge-Ampere 方程在更一般和更常见的几何对象中的应用，这些对象以前不知道存在联系。 通过寻找这个经典微分方程和几何之间的新关系，PI 旨在进一步了解可以存在什么样的几何结构。 此外，蒙日-安培方程通过相关的热流与几何形状密切相关。 预计这种热流将帮助我们理解一些有关代数方程解的老问题和难题。 事实上，代数方程的解定义了几何对象，热流使这些对象变形并可以从中提取信息。 PI 将研究 Monge-Ampere 热流在此类几何物体上的精确行为。