Kahler manifolds with curvature lower bound

OVERVIEW: The PI proposes to work on the structure of complete K¨ahler manifolds with curvature lower bound. The three topics below are interconnected with each other. The PI will study complete K¨ahler manifolds with nonnegative bisectional curvature. In this field, the main problem is the uniformization conjecture of Yau which states these manifolds are biholomorphic to the complex Euclidean space, if the bisectional curvature is positive. The PI will also study analytic and geometric properties of these manifolds. For instance, a conjecture of Yau states that Ricn is integrable on such manifolds. This would be a generalization of the Cohn Vossen inequality to higher dimensions. The PI will also study the Gromov-Hausdorff limits of K¨ahler manifolds with curvature lower bound. It is also natural to assume volume noncollapsing condition. The PI’s main interest is the degeneration of the complex structure. Also, there are interesting interactions between complex structure and metric structure. The PI also plans to study more function theory on K¨ahler manifolds with nonnegative curvature. Interesting questions include sharp dimension estimates and the existence of holomorphic functions with polynomial growth. INTELLECTUAL MERIT: The uniformization conjecture on positively curved K¨ahler manifolds is of fundamental importance in complex geometry. From the geometric point of view, in some sense, it generalizes the work of Riemann, Poincare and Koebe on the classification of simply connected Riemann surfaces. Better understanding of these manifolds can only be beneficial. The Gromov-Hausdorff convergence theory is an extremely powerful tool to study manifolds with curvature lower bound. This theory studies the regularity of limits of manifolds. Then one is able to retract nontrivial information for the original manifolds. The study of harmonic (holomorphic) functions was pioneered by Yau in 1970s. Since then this becomes an extremely active area. These objects provide a lot of connections between analysis, geometry and topology on manifolds. BROADER IMPACTS: In the past three years, the PI has taught several summer graduate courses in China. He had the opportunity to disseminate the foundations and research works to both graduate and undergraduate students. He is planning to continue teaching such topic courses in the future. Currently the PI is co-organizing the informal geometric analysis seminar at Northwestern University with Valentino Tosatti and Ben Weinkove. The academic atmosphere is strong but informal. Many fresh graduate students and undergraduate students have participated in the seminar. We benefited from each other during the discussions.