Holomorphic functions and some geometric problems on certain Kahler manifolds

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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.