Runge and Mergelyan theorems on families of open Riemann surfaces

TL;DR

This paper extends classical approximation theorems to families of open Riemann surfaces with varying complex structures, enabling the construction of continuous or smooth families of holomorphic maps to Oka manifolds and applications to minimal immersions.

Contribution

It develops Runge and Mergelyan approximation theorems for families of open Riemann surfaces with varying complex structures, and constructs families of holomorphic maps to Oka manifolds.

Findings

Established Runge and Mergelyan approximation theorems for families of Riemann surfaces.

Constructed continuous or smooth families of holomorphic maps to Oka manifolds.

Applied results to families of minimal and holomorphic immersions.

Abstract

Given a smooth open oriented surface $X$ endowed with a family of complex structures $\{J_b\}_{b\in B}$ of some H\"older class and depending continuously or smoothly on the parameter $b$ in a suitable topological space $B$, we construct continuous or smooth families $F_b:X\to Y$, $b\in B$, of $J_b$-holomorphic maps to any Oka manifold $Y$, with approximation on a suitable family of compact Runge sets in $X$. Along the way, we prove Runge and Mergelyan approximation theorems and Weierstrass interpolation theorem for functions on such families. We include applications to the construction of families of directed holomorphic immersions and conformal minimal immersions to Euclidean spaces.