# The stars at infinity in several complex variables

## TL;DR

This paper explores the asymptotic geometry of complex domains and their boundaries, discussing applications to complex dynamics, biholomorphisms, and K"ahler metrics, with some new results and open questions.

## Contribution

It introduces new perspectives on boundary behavior in complex geometry and formulates open problems in asymptotic and boundary analysis of complex domains.

## Key findings

- Unrecorded results on boundary estimates and asymptotic geometry.
- Formulation of new questions in complex dynamics and K"ahler geometry.
- Insights into boundary extensions of biholomorphisms.

## Abstract

This text reviews certain notions in metric geometry that may have further applications to problems in complex geometry and holomorphic dynamics in several variables. The discussion contains a few unrecorded results and formulates a number of questions related to the asymptotic geometry and boundary estimates of bounded complex domains, boundary extensions of biholomorphisms, the dynamics of holomorphic self-maps, Teichm\"uller theory, and the existence of constant scalar curvature metrics on compact K\"ahler manifolds.