Carleman Approximation of Maps into Oka Manifolds

TL;DR

This paper proves a Carleman approximation theorem for maps from Stein manifolds to Oka manifolds, allowing smooth maps satisfying Cauchy-Riemann equations to be approximated by holomorphic maps under certain convexity conditions.

Contribution

It establishes new Carleman approximation results for maps into Oka manifolds from Stein manifolds, extending approximation techniques under complex analytic conditions.

Findings

Approximation of smooth maps by holomorphic maps under convexity conditions.

Extension of Carleman approximation to maps into Oka manifolds.

Approximation results near totally real sets and convex sets.

Abstract

In this paper we obtain a Carleman approximation theorem for maps from Stein manifolds to Oka manifolds. More precisely, we show that under suitable complex analytic conditions on a totally real set $ M $ of a Stein manifold $X$, every smooth map $ X \rightarrow Y $ to an Oka manifold $Y$ satisfying the Cauchy-Riemann equations along $ M $ up to order $ k $ can be $ \mathscr{C}^k $-Carleman approximated by holomorphic maps $ X \rightarrow Y $. Moreover, if $ K $ is a compact $ \mathscr{O}(X) $-convex set such that $ K \cup M $ is $ \mathscr{O}(X) $-convex, then we can $ \mathscr{C}^k $-Carleman approximate maps which satisfy the Cauchy-Riemann equations up to order $ k $ along $ M $ and are holomorphic on a neighbourhood of $ K $, or merely in the interior of $K$ if the latter set is the closure of a strongly pseudoconvex domain.