Canonical Metrics, Geometric Flows and Formation of Singularities

规范度量、几何流和奇点的形成

• 批准号: 1406124
• 负责人: Jian Song
• 金额: $ 15.58万
• 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: 2014
• 资助国家: 美国
• 起止时间: 2014-08-01 至 2018-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1406124&HistoricalAwards=false
• 关键词: Canonical; Metrics; Geometric; Flows; Formation

Recent progress and influx of new ideas have unraveled a deep, rich and unifying structure among analysis, Riemannian geometry, pluripotential theory, classical several complex variables and the minimal model program in algebraic geometry. The proposed research work focuses on a number of open problems and developing programs on canonical metrics, geometric flows and complex Monge-Ampere equations arising from geometry and physics. The proposed project also aims to bring in research and teaching innovation in mathematics from various disciplines and have an immediate beneficial effect on graduate and undergraduate students at Rutgers as well as in the regional community of mathematics. The PI will also organize and participate in the integrated research/education programs and activities that will promote the education level of the nation. Furthermore, the PI plans to disseminate the exciting frontier research at the interface of geometry, analysis and algebra to a broad audience through lectures and survey papers. The PI will investigate and continue to make progress in the analytic minimal model program with Ricci flow. In particular, the PI will study both the finite time and long time formation of singularities of the Kahler-Ricci flow on algebraic varieties. Such singularity formation is reflected by canonical geometric/analytic surgeries equivalent to birational transformations and should be understood through global and local metric uniformization. The PI will also investigate the canonical metrics of Einstein type on singular varieties, in particular, the Riemannian geometric properties of such singular metrics and related moduli problems with applications in string theory such as geometric transitions and mirror symmetry. The PI willy employ new theories and techniques from L^2-theory, nonlinear PDEs, Perelman's works and Cheeger-Colding theory. The outcome of the proposed research will develop new tools and give profound insights of the structure of the universe as well as many other applied sciences.

最近的进展和新思想的涌入，揭示了一个深刻的，丰富的和统一的结构之间的分析，黎曼几何，多能理论，经典的几个复杂的变量和最小模型程序在代数几何。拟议的研究工作集中在一些开放的问题和规范度量，几何流和复杂的蒙赫-安培方程所产生的几何和物理发展计划。拟议的项目还旨在从各个学科引入数学研究和教学创新，并对罗格斯大学的研究生和本科生以及区域数学社区产生直接的有益影响。PI还将组织和参与将提高国家教育水平的综合研究/教育方案和活动。此外，PI计划通过讲座和调查论文向广大受众传播几何，分析和代数界面上令人兴奋的前沿研究。PI将研究并继续在Ricci流的分析最小模型程序中取得进展。特别是，PI将研究有限时间和长时间形成的Kahler-Ricci流的奇异代数簇。这种奇异性的形成反映了典型的几何/分析手术相当于双有理变换，应该通过全球和当地的度量均匀化理解。PI还将研究奇异簇上的爱因斯坦型正则度量，特别是这种奇异度量的黎曼几何性质和相关的模问题，以及在弦理论中的应用，如几何跃迁和镜像对称。PI将采用L^2-理论、非线性偏微分方程、Perelman的工作和Cheeger-Colding理论中的新理论和新技术。拟议研究的结果将开发新的工具，并对宇宙结构以及许多其他应用科学提供深刻的见解。

项目成果

The Ricci flow on the sphere with marked points

• DOI: 10.4310/jdg/1577502023
• 发表时间: 2014-07
• 期刊: Journal of Differential Geometry
• 影响因子: 2.5
• 作者: D. Phong;Jian Song;J. Sturm;Xiaowei Wang