國立清華大學 數學系

Abstract In this talk, I will talk about a nonlinear partial differential equation so called the mean field equation originated from conformal geometry. In some critical case, the issue of existence of solution or not is highly nontrivial. It turns out that it is dependent on the monodromy structure of a complex linear ordinary differential equation so called the Lame equation. I will give a criterion of existence of solution to the mean filed equation through the study of monodromy group of the Lame equation. A generalized case will be related to stationary elliptic KdV solution which will also be mentioned in this talk.