Kahler Manifolds with Curvature Lower Bound

Northwestern University, Evanston IL

Investigators

Abstract

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.