Geometric Flows and Four-dimensional Geometry

本提案中的项目旨在加深对某些自然几何演化方程的理解，这些方程概括了Ricci流，重点是理解四维几何的各个方面。一个例子是曲率张量平方范数的梯度流，这是一个自然的四阶抛物型方程，其临界点统一了四维流形上的各种重要度量。另一个重要的例子是在Pi和G.Tian以前的工作中引入的将Kahler Ricci流推广到非Kahler复流形上。通过使用成熟的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，以理解这些流的奇点形成。除了理解物理自然方程的内在价值外，一个可能的直接应用是理解复杂曲面的拓扑。几何流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，人们可以对拓扑结构有更深的理解。此外，这些方程通常具有物理动机，因此，通过理解它们，我们可以洞察自然的物理过程。即使在这些理论用途之外，几何流最近也出现了工业应用。因此，拟议的研究将增加我们对这些重要领域的总体了解。

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.