Different curvature flows and their long time behaviour

不同曲率流及其长期行为

• 批准号: 1110145
• 负责人: Natasa Sesum
• 金额: $ 11.63万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-06-30 至 2014-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1110145&HistoricalAwards=false
• 关键词: Different; curvature; flows; their; long

The proposer is interested in a long time behaviour of different parabolic flows, such as the Ricci flow, the Yamabe flow and different curvature flows of hypersurfaces in the euclidean space. More precisely, the proposer would like to understand the structure of possible singular limiting metrics one gets. Since the ancient solutions occur as singularity models of finite time singularities, the proposer suggests to study the properties and the classification of those in the case of different flows. One special case of ancient solutions are the gradient shrinking solitons. There is much to be understood about their geometric properties especially in the complete higher dimensional cases which can help the classification of those. Related to the singularities I the proposer also suggests studying the optimal conditions under which one can guarantee the existence of a smooth solution to e.g. the Ricci flow and the mean curvature flow.The proposer is interested in studying different parabolic geometric flows since their parabolic properties tend to improve the properties of the initial geometric objects. For example, under certain conditions on the initial metric the Ricci flow tends to exist forever and converges to a metric of constant sectional curvature which tells us a lot about the topology of our manifold. That means one can sometimes use the parabolic geometric flows in order to resolve some issues in other mathematical fields. Ancient solutions are the solutions that come from all the way from negative infinity. The physicists are interested in understanding those solutions to the Ricci flow.

提出者对不同抛物线流的长期行为感兴趣，例如 Ricci 流、Yamabe 流和欧几里得空间中超曲面的不同曲率流。更准确地说，提议者希望了解可能获得的单一限制指标的结构。由于古代的解决方案是作为有限时间奇点的奇点模型出现的，因此提议者建议研究不同流情况下的属性和分类。古代解决方案的一种特殊情况是梯度收缩孤子。关于它们的几何特性有很多需要理解，特别是在完整​​的高维情况下，这可以帮助对它们进行分类。与奇点 I 相关，提议者还建议研究最佳条件，在该条件下可以保证存在平滑解决方案，例如里奇流和平均曲率流。提议者有兴趣研究不同的抛物线几何流，因为它们的抛物线特性往往会改善初始几何对象的特性。例如，在初始度量的某些条件下，里奇流倾向于永远存在并收敛到恒定截面曲率的度量，这告诉我们很多有关流形拓扑的信息。这意味着有时可以使用抛物线几何流来解决其他数学领域的一些问题。古代的解是从负无穷一路而来的解。物理学家有兴趣了解里奇流的这些解决方案。