Research on holomorphic mappings of Riemann surfaces----roles of handles played in the existence problem of holomorphic mappings

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A Riemann surface is a connected complex manifold of dimension one, which is topologically an orientable surface and hence has several handles in general. A Riemann surface homeomorphic to the surface obtained from a torus by removing one point is called a once-holed torus. The once-holed tori make a space of real dimension three. In the present research we fix a Riemann surface Y with a marked handle. We investigate the set A of once-holed tori X for which there is a holomorphic mapping X into Y whose image corresponds to the marked handle, and compare A with the subset consisting of once-holed tori X which allow holomorphic mappings of finite degree into Y. It turns out that A is the closure of B and that B is a proper subset of A unless Y is a torus or a once-holed torus.