Holomorphic mappings of Riemann surfaces with handles

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A Riemann surface is a surface at every point of which angles are defined. It is obtained from tori with holes together with a plane domain by identifying boundaries appropriately. The number of tori is called the genus of the Riemann surface. If you want to embed a Riemann surface R of genus one into a general Riemann surface S, that is, if you want to draw a map of R on S, then you will need some "space" on S. In the present research we introduce a method of measuring the space and establish a theorem analogous to the Koebe one-quarter theorem, a classical theorem in function theory on plane domains.