Problems on the geometric function theory in several complex variables and complex geometry

Abstract

OF THE DISSERTATION Problems on the geometric function theory in several complex variables and complex geometry by Yuan Yuan Dissertation Director: Prof. Xiaojun Huang(advisor) and Prof. Jian Song(co-advisor) The thesis consists of two parts. In the first part, we study the rigidity for the local holomorphic isometric embeddings. On the one hand, we prove the total geodesy for the local holomorphic conformal embedding from the unit ball of complex dimension at least 2 to the product of unit balls and hence the rigidity for the local holomorphic isometry is the natural corollary. Before obtaining the total geodesy, the algebraic extension theorem is derived following the idea in [MN] by considering the sphere bundle of the source and target domains. When conformal factors are not constant, we twist the sphere bundle to gain the pseudoconvexity. Then the algebraicity follows from the algebraicity theorem of Huang in the CR geometry. Different from the argument in the earlier works, the total geodesy of each factor does not directly follow from the properness because the codimension is arbitrary. By analyzing the real analytic subvariety carefully, we conclude that the factor is either a proper holomorphic rational map or a constant map. Lastly the total geodesy follows from a linearity criterion of Huang. On the other hand, we also derive the total geodesy for the local holomorphic isometries from the projective space to the product of projective spaces. In the second part, we give a proof for the convergence of a modified Kahler-Ricci