报告题目 (Title)：A Maximal Rank Theorem for the Homogenous Complex Monge-Ampere Equation
中文标题: 齐次Monge-Ampere方程的最大秩定理
报告人 (Speaker)：胡京辰博士（中科院华罗庚中心）
报告时间 (Time)：2023年4月28日(周五) 13:30-14:30
报告地点 (Place)：校本部C121
邀请人(Inviter)：席东盟、李晋、张德凯
报告摘要：The convexity of solutions for various PDE has been studied extensively. Probably, the earliest result is Caratheodory's proof showing the Green functions of the Laplace operator of a convex domain in the complex plane have convex level sets; then, over the past decades, the result has been generalized to solutions for general dimensional Laplacian and the pluricomplex Green's function, and in many situations the optimal convexity estimates have been proved.
In this talk, we will show level sets of solutions to the homogenous complex Monge-Ampere equation in a linearly convex ring-shaped domain are linearly convex; as a consequence, the Hessian of the solution, which is an n by n matrix has maximum rank n-1.