A study of spirallike domains: polynomial convexity, Loewner chains and dense holomorphic curves

TL;DR

This paper investigates the polynomial convexity of spirallike domains, characterizes conditions for embedding univalent functions into Loewner chains, and demonstrates the existence of universal holomorphic maps related to hypercyclicity in complex analysis.

Contribution

It establishes polynomial convexity for spirallike domains, provides a criterion for embedding functions into Loewner chains, and constructs universal holomorphic maps with applications to hypercyclicity.

Findings

Closure of spirallike domains is polynomially convex.

Necessary and sufficient conditions for embedding into Loewner chains.

Existence of universal holomorphic maps for any bounded pseudoconvex spirallike domain.

Abstract

In this paper, we prove that the closure of a bounded pseudoconvex domain, which is spirallike with respect to a globally asymptotic stable holomorphic vector field, is polynomially convex. We also provide a necessary and sufficient condition, in terms of polynomial convexity, on a univalent function defined on a strongly convex domain for embedding it into a filtering Loewner chain. Next, we provide an application of our first result. We show that for any bounded pseudoconvex strictly spirallike domain $\Omega$ in $\mathbb{C}^n$ and given any connected complex manifold $Y$, there exists a holomorphic map from the unit disc to the space of all holomorphic maps from $\Omega$ to $Y$. This also yields us the existence of $\mathcal{O}(\Omega, Y)$-universal map for any generalized translation on $\Omega$, which, in turn, is connected to the hypercyclicity of certain composition operators on the space of manifold valued holomorphic maps.