Canonical Metrics, the Kahler-Ricci Flow, and Their Applica1ons

规范度量、Kahler-Ricci 流及其应用

• 批准号: 1711439
• 负责人: Jian Song
• 金额: $ 19.21万
• 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: 2017
• 资助国家: 美国
• 起止时间: 2017-09-01 至 2022-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1711439&HistoricalAwards=false
• 关键词: Canonical; Metrics; Kahler; Ricci; Flow

The proposed research work focuses on a number of open problems and developing programs on canonical metrics in Kahler geometry, geometric flows, and complex Monge-Ampere equations in relation to geometry and physics. Recent progress and influx of new ideas have unraveled a deep, rich, and unifying structure among analysis, partial differential equations, complex Riemannian geometry, and algebraic geometry. The project also aims to bring in research and teaching innovation in mathematics from various disciplines and have an immediate beneficial effect on undergraduate and graduate students at Rutgers as well as in the regional mathematical community. The principal investigator will continue to organize and participate in the integrated research/education programs and activities that will promote the education level of the nation. Furthermore, the principal investigator plans to disseminate the exciting frontier research at the interface of analysis and geometry to a broad audience through lectures and survey papers.These projects will investigate canonical metrics of Einstein type on Kahler varieties with mild singularities. In particular, the principal investigator will study the Riemannian geometric properties of such singular metrics and analytic moduli problems for Kahler-Einstein manifolds. The PI will continue to make progress in the analytic minimal model program with Ricci flow by studying 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 surgeries equivalent to birational transformations and should be understood through global and local metric uniformization. The PI willy employ new theories and techniques from L^2-theory, nonlinear PDEs and Cheeger-Colding theory. The research will develop new tools and give profound insights and understanding of topological, geometric and algebraic structures of complex spaces.

拟议的研究工作着重于许多开放问题，并开发有关Kahler几何形状，几何流量和复杂的Monge-Ampere方程的规范指标的程序。新思想的最新进展和涌入已经揭示了分析，部分微分方程，复杂的Riemannian几何形状和代数几何形状之间的深层，丰富和统一的结构。该项目还旨在引入来自各个学科的数学研究和教学创新，并对Rutgers以及区域数学社区的本科生和研究生产生直接有益的影响。首席研究人员将继续组织和参加将促进国家教育水平的综合研究/教育计划和活动。此外，首席研究人员计划通过讲座和调查论文在分析和几何学的界面上传播令人兴奋的边界研究。这些项目将调查爱因斯坦类型的规范指标，这些指标涉及具有轻度奇异性的Kahler品种。特别是，主要研究者将研究Kahler-Instein歧管的这种奇异指标和分析模量问题的Riemannian几何特性。 PI将通过研究代数品种上Kahler-Ricci流的奇异性的有限时间和长时间形成，并在RICCI流中继续取得进展。这种奇异性的形成反映了相当于异性转化的规范几何手术，应通过全球和局部度量均匀化来理解。 Pi Willy采用了L^2理论，非线性PDE和Cheeger-Colding理论的新理论和技术。 该研究将开发新的工具，并为复杂空间的拓扑，几何和代数结构提供深刻的见解和理解。

项目成果

Metric rigidity of Kahler manifolds with lower Ricci bounds and almost maximal volume.

具有下里奇界和几乎最大体积的卡勒流形的公制刚性。

• 发表时间: 2021
• 期刊: Proceedings of the American Mathematical Society
• 影响因子: 1
• 作者: Ved Datar, Harish Seshadri

Schauder estimates for equations with cone metrics, I

具有圆锥度量的方程的 Schauder 估计，I

• 发表时间: 2021
• 期刊: Indiana University mathematics journal
• 影响因子: 1.1
• 作者: Bin Guo, Jian Song