The Ricci flow has become an important tool to search classical metrics on manifolds since it first appeared in Hamilton's seminal 1982 paper. As an important evolutionary equation, it sets up a bridge between geometry and topology. In the past three decades, there have been many exciting achievements of the Ricci flow. In 2002, Perelman used the Ricci flow to solve the folklore Poincare conjecture. In 2007, Richard Schoen and Simon Brendle used the Ricci flow to prove the famous sphere theorem. These examples and many others highlight one fact that the Ricci flow is a powerful tool which deserves intensive study. The success of these previous examples is based on the knowledge of the global behavior of the Ricci flows with special conditions. Specially, either the dimension of the underlying manifold is three, or the curvature operator (or isotropic curvature) is nonnegative. However, in a general higher dimensional Ricci flow, we can hardly determine the sign of the curvature operator. The global picture of the Ricci flow is still unclear. There remain a lot of technical difficulties to overcome. Therefore, the study of the Ricci flows with weaker curvature constraints becomes natural and necessary. The Ricci flows' behavior under sectional curvature and Ricci curvature bounds have been solved by Hamilton and Sesum. Naturally, the next step is to understand the behavior of the Ricci flow under the condition that scalar curvature is uniformly bounded. On the other hand, Perelman's fundamental work reveals that there are many Ricci flows where scalar curvature is uniformly bounded. Therefore, the Ricci flows with bounded scalar curvature deserve comprehensive study. My research proposal is to study these Ricci flows.
The Ricci flow is an evolution equation solution on a Riemannian manifold. The Ricci flow is an important tool to find Einstein metrics, which are crucial in general relativity and mirror symmetry, my study is closely related to physics and Kahler geometry. It naturally interacts with mathematical physics, algebraic geometry, algebraic topology, complex analysis and partial differential equations. Therefore, the study of the Ricci flow has broader impact outside the area of geometric analysis. Among all Ricci flows, the Ricci flow with bounded scalar curvature is a very important type. This type of Ricci flows appear naturally in many settings. For example, according to the deep work of Perelman, the scalar curvature is uniformly bounded along the Ricci flows on many Kahler manifolds. My research proposal focuses on the study of the Ricci flows with bounded scalar curvature. The success of this project will greatly benefit the understanding of properties of many Riemannian manifolds.
Dr. Perelman solved the Poincare conjecture in 2002 and won the fields medal for his work in 2006. His success was based on his precise analysis of singularity formed by the Ricci flow on 3-dimensional manifolds. One naturally expects that his results can be generalized to high dimensional Ricci flow. It turns out to be very hard. One need to assume futher conditions for the study of singularities. We assume the condition that scalar curvature is uniformly bounded. This condition is natural since it is satisfied by many high-dimensional Ricci flows. For example, the Kahler Ricci flows on Fano manifolds. I made break through in the study of such Ricci flows under the general support of NSF1006518, NSF-1221330, and NSF-1312836. As a result, together with my colloaborators, we proved the famous Hamilton-Tian conjecture, which was open for about 20 years. This conjecture stays at the intersection of geometric flows, metric geometry, algebraic geometry and complex Monge-Ampere equation. In the process to prove this conjecture, we also used the fundamental idea from quantum mechanics and probability. Another important progress in generalizing the work of Perelman is to study the local behavior of Ricci flows, under the integral condition of scalar curvature. We obtain some powerful estimates, which are of fundamental importance in the study of Kahler geometry.
