Once-holed tori embedded in Riemann surfaces

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.