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.

提议者对不同抛物线流的长时间行为感兴趣，例如RICCI流动，Yamabe流量和欧几里得空间中高空曲面的不同曲率流动。更确切地说，提议者想了解一个人获得的可能奇异限制指标的结构。由于古代解决方案是作为有限时间奇点的奇异性模型出现的，因此建议者建议研究不同流量的特性和分类。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，特别是在完全可以帮助分类的较高维度的情况下，有很多理解的。与奇异性I相关的建议者还建议研究最佳条件，在这些条件下，人们可以保证存在平滑的解决方案，例如RICCI流量和平均曲率流。提议者有兴趣研究不同的抛物线几何流动，因为它们的抛物线特性倾向于改善初始几何对象的性质。例如，在初始度量标准的某些条件下，RICCI流倾向于永远存在，并收敛到恒定的截面曲率的度量标准，这向我们讲述了很多有关歧管的拓扑结构。这意味着有时可以使用抛物线几何流量来解决其他数学领域的某些问题。古代解决方案是从负面无穷大。物理学家有兴趣了解RICCI流动的解决方案。