On nonlinear Rudin-Carleson type theorem

TL;DR

This paper extends the classical Rudin-Carleson interpolation theorem to nonlinear settings, proving that continuous maps into complex manifolds defined on measure-zero subsets of the circle can be holomorphically extended to the disk.

Contribution

It introduces a nonlinear version of the Rudin-Carleson theorem for maps into complex manifolds, including cases with boundary conditions and interior constraints.

Findings

Existence of holomorphic extensions for continuous manifold-valued maps on measure-zero sets.

Extension results for maps into manifolds with boundary, respecting interior and boundary conditions.

Generalization of classical interpolation to nonlinear and manifold-valued functions.

Abstract

In this paper we study nonlinear interpolation problems for interpolation and peak-interpolation sets of function algebras. The subject goes back to the classical Rudin-Carleson interpolation theorem. In particular, we prove the following nonlinear version of this theorem: Let $\bar{\mathbb D}\subset \mathbb C$ be the closed unit disk, $\mathbb T\subset\bar{\mathbb D}$ the unit circle, $S\subset\mathbb T$ a closed subset of Lebesgue measure zero and $M$ a connected complex manifold. Then for every continuous $M$-valued map $f$ on $S$ there exists a continuous $M$-valued map $g$ on $\bar{\mathbb D}$ holomorphic on its interior such that $g|_S=f$. We also consider similar interpolation problems for continuous maps $f: S\rightarrow\bar M$, where $\bar M$ is a complex manifold with boundary $\partial M$ and interior $M$. Assuming that $f(S)\cap\partial M\ne\emptyset$ we are looking for holomorphic extensions $g$ of $f$ such that $g(\bar{\mathbb D}\setminus S)\subset M$.