Once-holed tori embedded in Riemann surfaces

Abstract

Once-holed tori are the most primitive noncompact Riemann surfaces of positive genus, and consitute a partially ordered set, the order being defined in terms of conforaml embeddings. We consider some families of once-holed tori that are conformally embedded in target Riemann surfaces of conformal mappings of a given noncompact Riemann surface of genus one, and establish an analogue of the one-quarter theorem of Koebe. We also investigate families of once-holed tori conformally embedded in a Riemann surface of positive genus.

On sets of marked once-holed tori allowing holomorphic mappings into Riemann surfaces with marked handle

“…Also, we gave a characterization for the existence of conformal mappings of a once-holed torus into another explicitly in terms of finitely many extremal lengths (see [7]). In [8] we examined the set of once-holed tori that can be conformally embedded into a given Riemann surface of positive genus. For topologically finite surfaces Kahn-Pilgrim-Thurston [4] has recently given a characterization for the existence of conformal embeddings in terms of extremal lengths.…”

Corrections and complements to “Once-holed tori embedded in Riemann surfaces”

“…For this, we first proved that (C-II) holds for any marked noncompact Riemann surface (R 0 , χ 0 ) of genus one (see [4,Theorem 7 and Proposition 9]). We thought that (C-I) would follow trivially from (C-II).…”