Investigation of Ricci Flows with Bounded Scalar Curvature

自从里奇流首次出现在汉密尔顿1982年开创性的论文中以来，它已经成为在流形上搜索经典度量的重要工具。作为一个重要的进化方程，它在几何和拓扑之间架起了一座桥梁。在过去的三十年里，利玛窦流已经取得了许多令人兴奋的成就。2002年，佩雷尔曼用里奇流解决了民间流传的庞加莱猜想。2007年，Richard Schoen和Simon Brendle用里奇流证明了著名的球体定理。这些例子和其他许多例子突出了一个事实，即利玛窦流是一个值得深入研究的强大工具。前面这些例子的成功是基于对特殊条件下里奇流的全局行为的了解。特别地，要么底层流形的维数是3，要么曲率算子（或各向同性曲率）是非负的。然而，在一般的高维Ricci流中，我们很难确定曲率算子的符号。利玛窦流的全球图景仍不清楚。还有许多技术上的困难需要克服。因此，研究具有较弱曲率约束的Ricci流是很自然和必要的。用Hamilton和Sesum求解了截面曲率和曲率边界下的Ricci流动特性。自然，下一步是了解标量曲率均匀有界条件下的Ricci流的行为。另一方面，佩雷尔曼的基本工作揭示了有许多标量曲率均匀有界的里奇流。因此，有界标量曲率的里奇流值得全面研究。我的研究计划是研究这些利玛窦流。里奇流是黎曼流形上演化方程的解。里奇流是发现爱因斯坦度量的重要工具，这在广义相对论和镜像对称中是至关重要的，我的研究与物理学和Kahler几何密切相关。它自然地与数学物理、代数几何、代数拓扑、复分析和偏微分方程相互作用。因此，里奇流的研究在几何分析领域之外具有更广泛的影响。在所有的里奇流中，有界标量曲率的里奇流是一种非常重要的类型。这种类型的利玛窦流自然地出现在许多环境中。例如，根据Perelman的深入研究，标量曲率沿许多Kahler流形上的Ricci流是均匀有界的。我的研究计划集中在有界标量曲率的Ricci流的研究上。该项目的成功将极大地促进对许多黎曼流形性质的理解。

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.