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

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.