An extension of H. Cartan’s theorem

Abstract:

In this article we prove that if $D\subset \mathbb {C}^n$, $n\ge 2$, is a bounded pseudoconvex domain with real analytic boundary, then for each $g(z)\in \mathrm {Aut}(D)$, there exists a fixed open neighborhood $\Omega _g$ of $\overline {D}$ and an open neighborhood $V_g$ of $g(z)$ in $\mathrm {Aut}(D)$ such that any $h(z)\in V_g$ can be extended holomorphically to $\Omega _g$, and that the action defined by \begin{align*} \pi :& V_g\times \Omega _g\to \mathbb {C}^n\ &(f,z)\mapsto \pi (f,z)=f(z) \end{align*} is real analytic in joint variables. This extends H. Cartan’s theorem beyond the boundary. Some applications are also discussed here.