Holomorphic Isometric Embeddings From The Unit Disk To Complex Manifolds

Holomorphic isometries between complex manifolds with respect to canonical Kaehler metrics are classical problems in complex differential geometry.The paper will discuss the problem of the holomorphic isometries from the unit disk to the complex manifold.The structure of the paper is as follows: Some definitions and theorems of complex manifolds are introduced in the chapter 1;the rigidity phenomena for holomorphic isometries from the complex unit ball to the product of complex unit balls is studied in Chapter 2;in Chapter 3,the holomorphic conformal embedding from the unit disk to complex manifolds will be studied,and basic properties such as existence and extension,and rigidity will be proved.The main results of the paper are following rigidity results:(1)Classification of holomorphic conformal embeddings from the unit disk to the polydisk with real analytic conformal functions that extend to an open neighborhood of the closed disk.(2)Classification of holomorphic conformal embeddings from the unit disk to the polydisk with real algebraic conformal functions that extend to an open neighborhood of the closed disk.(3)Classification of polynomial conformal embeddings from the unit disk to the product of two complex unit balls with real analytic conformal functions.