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  "body": "\n## Statement\n\nLet $D \\subseteq \\C^n$ be a connected open set that is a [[domain of holomorphy|4487CSC0MB]].\nLet $\\bar\\Delta$ be the closed unit disc in $\\C$, with boundary circle $\\partial\\Delta$.\nFor $j = 1, 2, \\dots$ let $\\varphi_j \\colon \\bar\\Delta \\to D$ be a continuous map that is holomorphic on the open disc.\nIf there is a [[compact set|0R6EWFKRCF]] $K \\subseteq D$ with $\\varphi_j(\\partial\\Delta) \\subseteq K$ for all $j$, then there is a compact set $L \\subseteq D$ with $\\varphi_j(\\bar\\Delta) \\subseteq L$ for all $j$.\n\n## Remarks\n\nLet $f$ be holomorphic on $D$.\nBy the maximum principle applied to $f \\circ \\varphi_j$, the values of $\\abs{f}$ on $\\varphi_j(\\bar\\Delta)$ are at most $\\max_K \\abs{f}$.\nSo every disc lies in the holomorphically convex hull $\\hat K_D$, which is compact by the [[Cartan–Thullen theorem|1XCBCB3MD7]].\nTake $L = \\hat K_D$.\n\nExample: the Hartogs figure $H = \\set{(z_1, z_2) : \\abs{z_1} \u003c 1, \\abs{z_2} \u003c 1, \\ \\abs{z_1} \u003c r \\text{ or } \\abs{z_2} \u003e s}$ fails the conclusion.\nFix $\\rho$ with $s \u003c \\rho \u003c 1$, choose $c_j$ with $\\abs{c_j} \u003c r$ and $c_j \\to c$ with $\\abs{c} = r$, and let $\\varphi_j(\\zeta) = (c_j, \\rho\\zeta)$.\nThese discs lie in $H$, and their boundaries lie in the compact set $\\set{\\abs{z_1} \\le r, \\abs{z_2} = \\rho} \\subseteq H$.\nBut the centres $(c_j, 0)$ tend to $(c, 0) \\notin H$, so no compact $L \\subseteq H$ contains all the discs.\nHence $H$ is not a domain of holomorphy.\n\nThe converse also holds: an open set with the disc property of the statement is pseudoconvex, and hence a domain of holomorphy by the solution of the Levi problem.\n\n## Explanations\n\n### L1 human\n\nThink of a sequence of holomorphic discs inside a domain whose rims stay in a fixed compact region.\nIn a domain of holomorphy the discs themselves cannot escape to the boundary either.\nIf a domain allows the discs to run out while their rims stay inside, then holomorphic functions get trapped and must extend.\n\n### L2 human\n\nThe continuity principle is one of the oldest forms of pseudoconvexity.\nIt is a convenient practical test.\nA single family of discs whose rims stay inside but whose centres leave the domain shows that the domain is not a domain of holomorphy.\nThe proof above shows how it follows from holomorphic convexity and the maximum principle.\n",
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