Analytic continuation in mapping spaces

TL;DR

This paper proves that holomorphic functions on certain mapping spaces from a compact manifold to a Stein manifold minus a compact set can be analytically continued to larger mapping spaces, extending classical ideas of analytic continuation.

Contribution

It establishes a new analytic continuation result for holomorphic functions on mapping spaces between Stein manifolds and their complements, generalizing classical complex analysis.

Findings

Holomorphic functions on $M'_S$ extend analytically to $M_S$.

Extension may be multivalued.

Applicable to mapping spaces from compact manifolds to Stein manifolds.

Abstract

We consider a Stein manifold $M$ of dimension $\geq 2$ and a compact subset $K\subset M$ such that $M'=M\backslash K$ is connected. Let $S$ be a compact differential manifold, and let $M_S$, resp. $M'_S$ stand for the complex manifold of maps $S\to M$, resp. $S\to M'$, of some specified regularity, that are homotopic to constant. We prove that any holomorphic function on $M'_S$ continues analytically to $M_S$ (perhaps as a multivalued function).