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，以推广利玛奇流，重点是对四维几何的理解。一个例子是曲率张量的平方范数的梯度流，这是一个自然的四阶抛物方程，它的临界点统一了四流形上各种重要的度量类。另一个重要的例子是将Kahler Ricci流推广到非Kahler复流形，这是PI和G. Tian在之前的工作中引入的。通过利用发达的利玛窦流理论的方法以及复杂几何的技术，PI建议完善他现有的工作，以了解这些流动的奇点形成。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表面的拓扑结构。几何流的方法是一种相对较新的理解几何物体结构的方法。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，我们可以深入了解自然物理过程。即使在这些理论应用之外，几何流最近也看到了工业应用。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.