Investigation of Ricci Flows with Bounded Scalar Curvature

自1982年汉密尔顿的开创性论文中首次出现以来，里奇流已经成为研究流形上经典度量的重要工具。 作为一个重要的演化方程，它在几何学和拓扑学之间架起了一座桥梁。在过去的三十年里，利玛窦流取得了许多令人兴奋的成就。2002年，Perelman利用Ricci流解决了庞加莱猜想。2007年，Richard Schoen和Simon Brendle使用Ricci流证明了著名的球面定理。这些例子和许多其他例子突出了一个事实，即利玛窦流是一个强大的工具，值得深入研究。前面这些例子的成功是基于对特殊条件下Ricci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（或迷向曲率）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。利玛窦流动的全球图景仍然不清楚。还有许多技术难题需要克服。因此，对具有较弱曲率约束的Ricci流的研究就变得自然和必要。汉密尔顿和Sesum已经解决了Ricci流在截面曲率和Ricci曲率边界下的行为。 自然地，下一步是理解在标量曲率一致有界的条件下里奇流的行为。另一方面，Perelman的基本工作表明，有许多Ricci流的标量曲率是一致有界的。因此，具有有界数量曲率的Ricci流值得深入研究。我的研究计划是研究这些Ricci流，Ricci流是黎曼流形上的发展方程解。Ricci流是寻找爱因斯坦度规的重要工具，而爱因斯坦度规在广义相对论和镜像对称中起着至关重要的作用，本文的研究与物理学和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，对Ricci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Ricci流中，具有有界数量曲率的Ricci流是一类非常重要的流。这种类型的利玛窦流自然出现在许多设置。例如，根据Perelman的深入工作，在许多Kahler流形上，标量曲率是沿着Ricci流一致有界的。我的研究计划主要集中在有界标量曲率的Ricci流的研究。该项目的成功将极大地有助于理解许多黎曼流形的性质。

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.