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  "body": "\n## Definition\n\nLet $D \\subseteq \\C^n$ be a domain, that is, a nonempty connected open set.\nA *holomorphic extension* of $D$ is a triple $(X, \\pi, \\alpha)$ with two properties.\n\n1. $(X, \\pi)$ is a [[Riemann domain over $\\C^n$|A15BF6QHDR]] and $\\alpha \\colon D \\to X$ is a [[holomorphic map|3MQTWKBBBT]] with $\\pi \\circ \\alpha = \\id_D$.\n2. For every [[holomorphic function|507C34W0ZT]] $f$ on $D$ there is a holomorphic function $F$ on $X$ with $F \\circ \\alpha = f$.\n\nAn *envelope of holomorphy* of $D$ is a holomorphic extension $(E, \\pi, \\alpha)$ of $D$ with the following property.\nFor every holomorphic extension $(X, \\rho, \\beta)$ of $D$ there is a holomorphic map $\\tau \\colon X \\to E$ with $\\pi \\circ \\tau = \\rho$ and $\\tau \\circ \\beta = \\alpha$.\n\n## Remarks\n\nThe function $F$ in clause 2 is unique, because $X$ is connected and $\\alpha(D)$ is a nonempty open subset of $X$.\nEvery domain $D$ has an envelope of holomorphy, and it is unique up to an isomorphism of Riemann domains compatible with the maps from $D$.\nIt can be built from the germs of holomorphic functions obtained from $D$ by analytic continuation.\n\nExample: for the [[Hartogs figure|GHJC53RFBT]] $H$ in $\\C^n$, the polydisc $\\Delta^n$ with the inclusion map is an envelope of holomorphy.\nEvery holomorphic function on $H$ extends to $\\Delta^n$; and $\\Delta^n$ is a domain of holomorphy, which gives the maximality.\n\nNon-example: for $D = \\set{z \\in \\C^2 : 0 \u003c \\abs{z} \u003c 1}$, the triple $(D, \\id, \\id)$ is a holomorphic extension but not an envelope.\nThe extension given by the ball $\\set{\\abs{z} \u003c 1}$ admits no map $\\tau$ into $D$ over $\\C^2$, since $\\tau$ would have to be the identity on the ball.\n\nA domain $D$ is a domain of holomorphy exactly when $(D, \\id, \\id)$ is its envelope of holomorphy.\n",
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