Investigation of Ricci Flows with Bounded Scalar Curvature

具有有界标量曲率的 Ricci 流研究

• 批准号: 1221330
• 负责人: Bing Wang
• 金额: $ 9.27万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-10-27 至 2012-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1221330&HistoricalAwards=false
• 关键词: Investigation; Ricci; Flows; Bounded; Scalar

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.

自从汉密尔顿（Hamilton）的1982年论文中首次出现以来，RICCI流量已成为搜索流形的经典指标的重要工具。 作为重要的进化方程，它在几何和拓扑之间建立了桥梁。在过去的三十年中，RICCI流动取得了许多令人兴奋的成就。 2002年，佩雷尔曼（Perelman）利用里奇（Ricci）的流程来解决民间传说中的猜想。 2007年，理查德·舒恩（Richard Schoen）和西蒙·布伦德尔（Simon Brendle）利用里奇（Ricci）流证明了著名的球体定理。这些例子和许多其他例子强调了Ricci流是一个应有深入研究的强大工具的一个事实。这些先前示例的成功是基于对RICCI流动的特殊条件的全球行为的了解。特别是，基础歧管的尺寸为三个，或者曲率算子（或各向同性曲率）是非负的。 但是，在一般较高的尺寸RICCI流中，我们几乎无法确定曲率操作员的符号。 RICCI流动的全球情况尚不清楚。仍有许多技术困难要克服。因此，对RICCI流动的曲率约束的研究变得自然而必要。汉密尔顿和Sesum解决了截面曲率和RICCI曲率界限下的Ricci流动行为。 自然，下一步是了解标量曲率统一边界的条件下RICCI流的行为。另一方面，Perelman的基本工作表明，有许多RICCI流动，标态曲率均匀界定。因此，RICCI以有限的标态曲率流动应进行全面的研究。我的研究建议是研究这些RICCI流动。Ricci流是Riemannian歧管上的进化方程解决方案。 RICCI流是找到爱因斯坦指标的重要工具，在一般相对论和镜面对称性中至关重要，我的研究与物理学和Kahler几何形状密切相关。 它自然与数学物理学，代数几何形状，代数拓扑，复杂分析和部分微分方程相互作用。 因此，对RICCI流量的研究在几何分析领域之外具有更大的影响。 在所有RICCI流中，具有界标曲率曲率的RICCI流是非常重要的类型。这种类型的Ricci流在许多情况下自然出现。例如，根据佩雷尔曼（Perelman）的深刻工作，标量曲率沿Ricci流动在许多Kahler歧管上均匀界定。我的研究建议着重于对有界标态曲率的RICCI流动的研究。该项目的成功将极大地有助于对许多Riemannian流形的财产的理解。