Envelope of holomorphy

Remarks

The function $F$ in clause 2 is unique, because $X$ is connected and $\alpha(D)$ is a nonempty open subset of $X$. Every domain $D$ has an envelope of holomorphy, and it is unique up to an isomorphism of Riemann domains compatible with the maps from $D$. It can be built from the germs of holomorphic functions obtained from $D$ by analytic continuation.

Example: for the Hartogs figure $H$ in $\C^n$, the polydisc $\Delta^n$ with the inclusion map is an envelope of holomorphy. Every holomorphic function on $H$ extends to $\Delta^n$; and $\Delta^n$ is a domain of holomorphy, which gives the maximality.

Non-example: for $D = \set{z \in \C^2 : 0 < \abs{z} < 1}$, the triple $(D, \id, \id)$ is a holomorphic extension but not an envelope. The extension given by the ball $\set{\abs{z} < 1}$ admits no map $\tau$ into $D$ over $\C^2$, since $\tau$ would have to be the identity on the ball.

A domain $D$ is a domain of holomorphy exactly when $(D, \id, \id)$ is its envelope of holomorphy.