Geometric Flows and Four-dimensional Geometry

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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.