Classification Problem of Holomorphic Isometries of the Unit Disk Into Polydisks

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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 Δ4. Moreover, we classify those holomorphic isometric embeddings with certain prescribed sheeting numbers. In addition, we prove that a known example in the space HIk(Δ,Δqk;q) is globally rigid for any integers k,q≥2, which generalizes Theorem 1.1 in [Ch16a].