Generalizations of Picard’s theorem for Riemann surfaces

Järvi, Pentti

doi:10.1090/S0002-9947-1991-1030508-9

Generalizations of Picard’s theorem for Riemann surfaces
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Trans. Amer. Math. Soc. 323 (1991), 749-763 Request permission

Abstract:

Let $D$ be a plane domain, $E \subset D$ a compact set of capacity zero, and $f$ a holomorphic mapping of $D\backslash E$ into a hyperbolic Riemann surface $W$. Then there is a Riemann surface $W’$ containing $W$ such that $f$ extends to a holomorphic mapping of $D$ into $W’$. The same conclusion holds if hyperbolicity is replaced by the assumption that the genus of $W$ be at least two. Furthermore, there is quite a general class of sets of positive capacity which are removable in the above sense for holomorphic mappings into Riemann surfaces of positive genus, except for tori.

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