Investigation of Ricci Flows with Bounded Scalar Curvature

自1982年在哈密尔顿开创性的论文中首次出现以来，Ricci流已经成为在流形上搜索经典度量的重要工具。作为一个重要的演化方程，它在几何和拓扑学之间架起了一座桥梁。在过去的三十年里，利玛窦取得了许多令人振奋的成就。2002年，佩雷尔曼利用利奇流解决了民间传说中的庞加莱猜想。2007年，Richard Schoen和Simon Brendle用Ricci流证明了著名的球面定理。这些例子和其他许多例子突出了一个事实，即利玛窦流动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具有特殊条件的Ricci流的全球行为的了解。特别地，要么底层流形的维度是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。利玛窦流动的全球图景仍不明朗。仍有许多技术困难需要克服。因此，研究具有较弱曲率约束的Ricci流就变得自然和必要了。在截面曲率和Ricci曲率边界下的Ricci流的行为已由Hamilton和Sesum求解。自然，下一步就是了解标量曲率一致有界条件下的Ricci流的行为。另一方面，Perelman的基本工作揭示了存在许多数量曲率一致有界的Ricci流。因此，有界标量曲率的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性中是至关重要的。我的研究与物理学和卡勒几何密切相关。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式相互作用。因此，对利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Ricci流中，有界标量曲率的Ricci流是一种非常重要的类型。这种类型的Ricci流在许多情况下都很自然地出现。例如，根据Perelman的深入工作，在许多Kahler流形上，标量曲率沿着Ricci流一致有界。我的研究建议集中于研究具有有界标量曲率的Ricci流。这一项目的成功将极大地促进人们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.