Analytic continuation in mapping spaces
  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). 
Laszlo Lempert