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

拟议的研究工作集中在一些开放的问题和规范度量Kahler几何，几何流，复杂的蒙赫-安培方程的几何和物理的发展计划。最近的进展和新思想的涌入已经揭开了一个深刻的，丰富的，统一的结构之间的分析，偏微分方程，复杂的黎曼几何和代数几何。该项目还旨在从各个学科引入数学研究和教学创新，并对罗格斯大学的本科生和研究生以及区域数学界产生直接的有益影响。主要研究者将继续组织和参与综合研究/教育计划和活动，以提高国家的教育水平。此外，首席研究员计划通过讲座和调查论文向广大受众传播分析和几何界面上令人兴奋的前沿研究。这些项目将研究具有轻度奇点的Kahler簇上的爱因斯坦型正则度量。特别是，首席研究员将研究黎曼几何性质的奇异度量和解析模问题的Kahler-Einstein流形。PI将继续通过研究代数簇上Kahler-Ricci流的有限时间和长时间奇点的形成，在Ricci流的分析最小模型程序中取得进展。这种奇异性的形成反映了典型的几何手术相当于双有理变换，并应通过全球和当地的度量均匀化理解。PI将采用L^2-理论、非线性偏微分方程和Cheeger-Colding理论等新的理论和技术。 该研究将开发新的工具，并对复杂空间的拓扑，几何和代数结构给予深刻的见解和理解。

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.