A Rudin-Carleson theorem with uniform approximation for manifold-valued maps

Abstract

Given a closed set $E\subset \partial \mathbb{D}$ of measure zero and a continuous function $\phi :E\to \mathbb{C}$, the classical Rudin-Carleson interpolation theorem states that there exists a continuous function $F:\overline{\mathbb{D}}\to \mathbb{C}$ that is holomorphic on $\mathbb{D}$ and satisfies $F{|}_E=\phi$. In this paper we obtain a generalisation of the Rudin-Carleson theorem for maps $\phi :E\to X$ into arbitrary connected complex manifolds $X$ that combines interpolation of $\phi$ on $E$ with uniform approximation on compact subsets of $\overline{\mathbb{D}}\setminus E$ of another given continuous map $f:\overline{\mathbb{D}}\to X$ that is holomorphic on $\mathbb{D}$. Under the further assumption that $X$ is an Oka manifold we obtain a corollary that combines Rudin-Carleson interpolation of a continuous map $\phi :\overline{\mathbb{D}}\to X$ on $E$ with Runge approximation of $\phi$ on a compact set $K\subset \mathbb{D}$ without any holes on which $\phi$ is holomorphic.