# Proper holomorphic self-maps of symmetric powers of balls

## TL;DR

This paper proves that proper holomorphic self-maps of symmetric powers of higher-dimensional balls are actually automorphisms derived from automorphisms of the original ball, revealing a strong structural rigidity.

## Contribution

It establishes a rigidity result for proper holomorphic self-maps of symmetric powers of balls, showing they are induced by automorphisms of the ball itself.

## Key findings

- Proper holomorphic self-maps are automorphisms
- Induced by automorphisms of the original ball
- Results hold for balls of dimension at least two

## Abstract

We show that each proper holomorphic self map of a symmetric power of the unit ball is an automorphism naturally induced by an automorphism of the unit ball, provided the ball is of dimension at least two.