Elliptic and parabolic complex Monge-Ampere equations on compact manifolds

PI计划调查三个领域的项目，与复杂的蒙日-安培方程的主题相关。第一个项目将建立在PI与tosati的工作基础上，tosati给出了紧化厄米流形的Yau定理的类比。将这些估计推广到复流形的bot - chen空间，得到了新的结果。第二个项目涉及代数变量上的Kahler-Ricci流，它可以被看作是抛物线复蒙日-安培方程。作为理解解析最小模型程序的一部分，PI和Song研究了代数变量上的Kahler-Ricci流，并给出了流收缩一个例外除数的必要条件。PI建议将这些结果扩展到处理更一般的奇点，并研究曲率张量沿Kahler-Ricci流的行为。最后的课题是关于辛流形上的Calabi-Yau方程。Donaldson推测Yau所解的复Monge-Ampere方程对于相容的几乎复杂结构的辛4流形具有类似的性质。他把他的猜想应用于辛拓扑。唐纳森猜想的特殊情况是由PI和他的合作者建立的。PI将对Calabi-Yau方程的放大集结构进行分析，其最终目标是证明Donaldson猜想。数学和物理学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。这个提议涉及到一个著名的、有几个世纪历史的数学对象——蒙日-安培方程。这个方程在几何学的研究中自然出现，并且与物理学中的爱因斯坦方程密切相关。这个项目的一个主要目标是找到蒙日-安培方程在更一般和更常见的几何物体上的应用，在这些物体上以前不知道存在联系。通过寻找经典微分方程和几何之间的新关系，PI旨在进一步了解什么样的几何结构可以存在。此外，蒙日-安培方程通过相关的热流与几何密切相关。期望这种热流将帮助我们理解一些古老而困难的代数方程解问题。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从中提取信息。PI将研究此类几何物体上蒙日-安培热流的精确行为。

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.