Runge embeddings, approximation of biholomorphisms on Stein manifolds, and the Loewner PDE

Abstract

We develop an extension-by-approximation principle for holomorphic Runge embeddings of increasing union of Stein manifolds into Stein manifolds with density property. The basic hypothesis is the existence, on each stage of the exhaustion, of a Runge isotopy which compresses the stage and whose terminal map extends holomorphically to the next stage. The resulting global embedding of the union may be chosen with Runge image, and every Runge embedding of a fixed stage can be approximated uniformly on compact subsets by the Runge embeddings of the union. We apply this principle to domains that are invariant under positive time part of holomorphic $(R,+)$-actions, to Stein manifolds carrying a semicomplete holomorphic vector field with globally attracting fixed point. It also gives a Runge embedding of $(\mathbb{C}^n\setminus \{z\in\mathbb{C}^n: f(z)=0\})\times \mathbb{C}$ in $\mathbb{C}^{n+1}$, which generalizes previous result of Runge embedding of $(\mathbb{C}^*)^n\times\mathbb{C}$ into $\mathbb{C}^{n+1}$. We also construct Stein globalization of an injective holomorphic semigroup action to holomorphic $(R,+)$-action. Finally, the abstract Loewner range of a Herglotz vector field is shown to admit a same-dimensional Runge embedding whenever the initial domain admits a Runge embedding into a Stein domain with density property; this yields a corresponding solution of the Loewner PDE with values in $\mathbb{C}^n$. We also give an example of non-Runge complete hyperbolic domain which admits $\mathbb{C}^n$-valued solution of the Loewner PDE.