Continuity principle for domains of holomorphy

Put $H:={\widehat{K}}_{\mathrm{\Omega }}$. By [L2], $H⋐\mathrm{\Omega }$. Let $S:=\{t\in [0,1]:{\mathrm{\Phi }}_t(\bar{\mathbb{D}})\subseteq \mathrm{\Omega }\}\mathrm{.}$ The initial-disc hypothesis gives $0\in S$. If $t\in S$, then the compact set ${\mathrm{\Phi }}_t(\bar{\mathbb{D}})$ lies in the open set $\mathrm{\Omega }$; uniform continuity of $\mathrm{\Phi }$ on the compact parameter product shows that the same containment holds for all parameters sufficiently close to $t$. Thus $S$ is open in $[0,1]$.

If $t\in S$ and $f\in \mathcal{O}(\mathrm{\Omega })$, then $f\circ {\mathrm{\Phi }}_t$ is holomorphic on $\mathbb{D}$ and continuous on its closure. By [L3] and the boundary hypothesis, $\mathrm{\mid }f({\mathrm{\Phi }}_t(\lambda ))\mathrm{\mid }\le {\sup }_{\mathrm{\partial }\mathbb{D}}\mathrm{\mid }f\circ {\mathrm{\Phi }}_t\mathrm{\mid }\le {\sup }_K\mathrm{\mid }f\mathrm{\mid }(\lambda \in \bar{\mathbb{D}})\mathrm{.}$ Since this holds for every $f$, one has ${\mathrm{\Phi }}_t(\bar{\mathbb{D}})\subseteq H$.

Let $t_n\in S$ and $t_n\to t$. For every $\lambda \in \bar{\mathbb{D}}$, step 2.1 gives ${\mathrm{\Phi }}_{t_n}(\lambda )\in H$. The compact set $H$ is closed in ${\mathbb{C}}^m$, so continuity of $\mathrm{\Phi }$ yields ${\mathrm{\Phi }}_t(\lambda )\in H\subseteq \mathrm{\Omega }$. Hence $t\in S$, and $S$ is closed. Since $[0,1]$ is connected and $S$ is nonempty, open, and closed, $S=[0,1]$.