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 -disk . Moreover, we classify those holomorphic isometric embeddings with certain prescribed sheeting numbers. In addition, we prove that a known example in the space is globally rigid for any integers , which generalizes Theorem 1.1 in [Ch16a].
"Classification Problem of Holomorphic Isometries of the Unit Disk Into Polydisks." Michigan Math. J. 66 (4) 745 - 767, November 2017. https://doi.org/10.1307/mmj/1505527453