Geometric Flows and Four-dimensional Geometry

本提案中的项目旨在加深对某些推广Ricci流的自然几何演化方程的理解，重点是理解四维几何的各个方面。 一个例子是曲率张量的平方范数的梯度流，一个自然的四阶抛物方程，其临界点统一了四流形上的各种重要度量。 另一个重要的例子是在PI和G的先前工作中引入的Kahler Ricci流到非Kahler复流形的推广。田 通过利用完善的Ricci流理论的方法以及复杂几何的技术，PI建议改进他现有的工作，以了解这些流的奇异性形成。 除了理解物理上自然的方程的内在价值外，一个可能的直接应用是理解复杂曲面的拓扑结构。几何流方法是理解几何对象结构的一种相对较新的技术。 最终，通过理解这些方程，人们可以深入了解拓扑结构。 此外，这些方程通常具有物理动机，因此通过理解它们，我们可以深入了解自然物理过程。 甚至除了这些理论上的用途外，几何流最近也得到了工业上的应用。 因此，拟议的研究将增加我们对这些重要领域的全面了解。

The projects in this proposal aim to deepen the understanding of certain natural geometric evolution equations generalizing the Ricci flow, with a focus on understanding aspects of four dimensional geometry. One example is the gradient flow of the square norm of the curvature tensor, a natural fourth-order parabolic equation whose critical points unify various important classes of metrics on four-manifolds. Another important example is a generalization of Kahler Ricci flow to non-Kahler complex manifolds introduced in prior work of the PI and G. Tian. By utilizing methods from the well-developed theory of Ricci flow as well as techniques from complex geometry, the PI proposes to refine his existing work to understand the singularity formation of these flows. Aside from the intrinsic value of understanding physically natural equations, one possible direct application is to understand the topology of complex surfaces.The method of geometric flows is a relatively new technique for understanding the structure of geometric objects. Ultimately, by understanding these equations one can gain a deep understanding of topological structures. Furthermore, these equations typically have a physical motivation, and hence by understanding them we gain insight into natural physical processes. Even beyond these theoretical uses geometric flows have recently seen industrial application. The proposed research will thus add to our overall understanding in these important areas.