Holomorphic functions and some geometric problems on certain Kahler manifolds

摘要奖：DMS 1406593，首席研究员：Gang LiuKahler流形是宇宙弦理论模型的基本组成部分。本项目研究Kahler流形的整体结构。 特别是，将解决Kahler流形的一致化的问题。这些定理推广了经典的单复变单值化定理。 在PI的项目中，数学的几个分支之间有许多相互作用，例如代数几何，分析，拓扑和微分几何。PI领域的更广泛应用包括分子结构、宇宙大尺度结构和液气边界。 PI建议在三个项目上工作，涉及Kahler流形上的函数理论和几何。第一个项目是研究与丘氏一致化猜想密切相关的问题。这些问题包括多项式增长的全纯函数环的有限生成，非负Ricci曲率流形上多项式增长的全纯函数的精确维数估计，以及Ni关于平均曲率衰减，最大体积增长和多项式增长的全纯函数的存在性之间等价性的一个猜想.在第二个项目中，PI将寻找具有非负标量曲率的Kahler度量的障碍物。PI还计划证明具有非正对分曲率的紧致Kahler流形的科代拉维数是同伦不变量。主要工具是PI的结构定理，这些流形。

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.