# Some aspects of holomorphic mappings: a survey

## TL;DR

This survey reviews key properties and methods related to proper holomorphic mappings between complex domains, highlighting geometric techniques, recent advances, and open problems in the field.

## Contribution

It provides a comprehensive overview of geometric methods and recent developments in the theory of proper holomorphic mappings, including proof sketches and open questions.

## Key findings

- Discussion of the Reflection Principle and scaling method
- Overview of the Kobayashi-Royden metric applications
- Summary of recent significant results in the field

## Abstract

This expository paper is concerned with the properties of proper holomorphic mappings between domains in complex affine spaces. We discuss some of the main geometric methods of this theory, such as the Reflection Principle, the scaling method, and the Kobayashi-Royden metric. We sketch the proofs of certain principal results and discuss some recent achievements. Several open problems are also stated.