Holomorphic isometries between products of complex unit balls

TL;DR

This paper explores holomorphic isometries involving the Poincaré disk, providing new examples and results, including an irrational, algebraic proper map from the disk to the complex unit ball, enriching the understanding of such mappings.

Contribution

It offers a detailed exposition on holomorphic isometries and introduces novel examples and results, notably an irrational algebraic proper map from the disk to the ball.

Findings

Explicit descriptions of holomorphic isometries from the Poincaré disk to polydisks

Construction of an irrational algebraic proper map from the disk to the ball

New theoretical results on holomorphic isometries

Abstract

We first give an exposition on holomorphic isometries from the Poincar\'e disk to polydisks and from the Poincar\'e disk to the product of the Poincar\'e disk with a complex unit ball. As an application, we provide an example of proper holomorphic map from the unit disk to the complex unit ball that is irrational, algebraic and holomorphic on a neighborhood of the closed unit disk. We also include some new results on holomorphic isometries.