Holomorphic functions and some geometric problems on certain Kahler manifolds

1. Project Description The PI proposes to work on three projects which involve function theory and geometry on K¨ahler manifolds. The PI will study problems which are closely related with the uniformization conjecture of Yau. These problems include the finite generation of the ring of holomorphic functions; the sharp dimension estimates for holomorphic functions with polynomial growth on manifolds with nonnegative Ricci curvature; the relations between the monotonicity of Ni and the three circle theorem by the PI. For open complex manifolds, the PI will seek obstructions to K¨ahler metrics with nonnegative scalar curvature. He will also work on obstructions to K¨ahler metrics with nonnegative scalar curvature outside compact sets. The PI also plans to show that the Kodaira dimension of compact K¨ahler manifolds with nonpositive bisectional curvature is a homotopy invariant. The main tool is the PI’s structure theorem for these manifolds. 2. Intellectual Merit The uniformization conjecture on positively curved manifolds is of fundamental importance in complex geometry. It generalizes the work of Riemann, Poincare and Koebe on the classification of simply connected Riemann surfaces. A better understanding on the function theory on these manifolds can only be beneficial. The relations between curvature, topology and complex structure are important in differential geometry and complex geometry. The proposed research will sharpen our understanding in the obstruction to certain curvature for a given complex structure. The third project aims at showing that Kodaira dimension is a homotopy invariant under some curvature assumption. This shall provide us more connections between algebraic geometry, topology and differential geometry. 3. Broader Impacts The PI’s subject of investigation belongs to differential geometry and global analysis. Differential geometry studies the geometry of manifolds. This turns out to be extremely useful in general relativity and string theory. The applications of the PI’s field include the structure of molecules, the large scale of the universe and the liquid gas boundary.