Kahler Manifolds with Curvature Lower Bound

复数在现代数学中无处不在，从求解二次方程，到模拟流体流动，再到探索隐藏在弦理论卷曲维度中的空间。复数值函数的自然域是复流形，包括n维复欧几里德空间。这个项目关注的是一类自然的复杂流形（称为Kahler流形）的几何形状与这些流形上的复杂函数的行为之间的关系。复分析（复值函数的研究）中最美丽的结果之一是均匀化定理，它说一维复流形本质上只有三种形状：球体、圆盘或平面，对应于正曲率、负曲率或零曲率。长期以来，数学家们一直没有找到高维统一化的版本；例如，我们推测如果一个开放的n维复流形具有正曲率，那么它就是复n空间的一个副本。PI将解决这个长期存在的猜想。由此产生的研究，位于许多数学分支的交叉点，将在分析，代数，微分几何和拓扑学之间建立新的有趣的联系。更准确地说，PI将研究Yau的均匀化猜想，该猜想指出具有正对分曲率的完全非紧Kahler流形对复n空间是生物全纯的。从20世纪80年代初的莫绍佑开始，迄今为止，人们对这一猜想进行了多次尝试。在讨论均匀化猜想的同时，还讨论了Siu关于具有正对分曲率的完全非紧Kahler流形的斯坦性猜想等相关问题。在PI的早期工作中，Gromov-Hausdorff收敛理论将作为研究这些猜想的重要工具。PI还将研究具有曲率下界的Kahler流形的Gromov-Hausdorff极限（例如，复杂结构的退化）。在某种意义上，这是Donaldson-Sun在Kahler流形上的突破结果的推广。期望的是极限空间应该带有一个自然的复解析结构。

Complex numbers are everywhere in modern mathematics, from solving quadratic equations, to modeling fluid flow, to probing the spaces hidden in the curled-up dimensions of string theory. The natural domains of complex-number-valued functions are complex manifolds, including the n-dimensional complex Euclidean space. This project concerns the relationship between the geometry of a natural class of complex manifolds, called Kahler manifolds, and the behavior of complex functions on these manifolds. One of the most beautiful results in complex analysis (the study of complex-valued functions) is the uniformization theorem, which says that one-dimensional complex manifolds have essentially only three shapes: a sphere, a disc, or a plane, corresponding to positive, negative, or zero curvature. A higher-dimensional version of uniformization has long eluded mathematicians; for example, it is conjectured that if an open n-dimensional complex manifold has positive curvature then it is a copy of complex n-space. The PI will address this long-standing conjecture. The resulting research, which lies at the intersection of many branches of mathematics, will establish new and interesting connections between analysis, algebra, differential geometry and topology.More precisely, the PI will work on the uniformization conjecture of Yau, which states that a complete noncompact Kahler manifold with positive bisectional curvature is biholomorphic to complex n-space. So far there have been numerous attempts at this conjecture, starting with Mok-Siu-Yau in early 1980s. Along with the uniformization conjecture, the PI will also address related problems such as Siu's conjecture on Stein-ness of complete noncompact Kahler manifolds with positive bisectional curvature. As in the PI's earlier work, the Gromov-Hausdorff convergence theory will serve as an important tool to study these conjectures. The PI will also study the Gromov-Hausdorff limit of Kahler manifolds with curvature lower bound (e.g., the degeneration of the complex structure). In some sense, this is a generalization of the breakthrough result of Donaldson-Sun on Kahler manifolds. The expectation is that the limit space should carry a natural complex analytic structure.