Holomorphic functions and some geometric problems on certain Kahler manifolds

全纯函数和某些卡勒流形上的一些几何问题

• 批准号: 1700852
• 负责人: Gang Liu
• 金额: $ 3.18万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-10-15 至 2018-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1700852&HistoricalAwards=false
• 关键词: Holomorphic; functions; some; geometric; problems

AbstractAward: DMS 1406593, Principal Investigator: Gang LiuKahler manifolds are a basic building block in the string theory model of the universe. This project studies the global structure of Kahler manifolds. In particular, conjectures on the uniformization of Kahler manifolds will be addressed. These conjectures generalize the classical uniformization theorem in one complex variable. In the PI's project, there are many interactions among several branches of mathematics, e.g. algebraic geometry, analysis, topology and differential geometry. Wider applications of the PI's field include the structure of molecules, the large scale structure of the universe and the liquid gas boundary. The PI proposes to work on three projects which involve function theory and geometry on Kahler manifolds. The first project is to study problems which are closely related to the uniformization conjecture of Yau. These problems include the finite generation of the ring of holomorphic functions of polynomial growth, sharp dimension estimates for holomorphic functions with polynomial growth on manifolds with nonnegative Ricci curvature, and a conjecture of Ni on the equivalence between average curvature decay, maximal volume growth and the existence of holomorphic functions of polynomial growth. In the second project, the PI will seek obstructions to Kahler metrics with nonnegative scalar curvature. The PI also plans to show that the Kodaira dimension of compact Kahler manifolds with nonpositive bisectional curvature is a homotopy invariant. The main tool is the PI's structure theorem for these manifolds.

Abstractaward：DMS 1406593，主要研究人员：Gang Liukahler歧管是宇宙弦理论模型中的基本构件。该项目研究了卡勒流形的全球结构。 特别是，将解决有关卡勒歧管均匀化的猜想。这些猜想在一个复合变量中概括了经典的统一定理。 在PI的项目中，数学的几个分支之间存在许多互动，例如代数几何，分析，拓扑和差异几何形状。 PI场的更广泛应用包括分子的结构，宇宙的大规模结构和液体气体边界。 PI建议在Kahler歧管上进行涉及功能理论和几何形状的三个项目。第一个项目是研究与Yau的统一化猜想密切相关的问题。这些问题包括多项式生长的圆锥形功能的有限生成，对多项式生长的尖锐尺寸估计在多项式生长上具有多项性RICCI曲率的流形，以及NI对NI在平均曲率衰减，最大体积增长和最大率增长之间的等价之间的猜想。在第二个项目中，PI将寻求对具有非负标态曲率的Kahler指标的障碍。 PI还计划表明，与非阳性双弯曲曲率的紧凑型Kahler歧管的Kodaira尺寸是同型不变的。主要工具是这些流形的PI结构定理。