Classification Problem of Holomorphic Isometries of the Unit Disk Into Polydisks

Abstract

We study the classification problem of holomorphic isometric embeddings of the unit disk into polydisks as in [Ng10, Ch16a]. We give a complete classification of all such holomorphic isometries when the target is the $4$-disk ${\Delta }^4$. Moreover, we classify those holomorphic isometric embeddings with certain prescribed sheeting numbers. In addition, we prove that a known example in the space ${\mathbf{HI}}_k(\Delta ,{\Delta }^{qk};q)$ is globally rigid for any integers $k,q\ge 2$, which generalizes Theorem 1.1 in [Ch16a].