HOLOMORPHIC MAPPINGS OF ONCE-HOLED TORI

Let T be the space of marked once-holed tori and Y-0 be a Riemann surface with marked handle. We investigate geometric properties of the set T-a[Y-0] of X is an element of T that allow holomorphic mappings of X into Y-0. We also examine the set T-c[Y-0] of marked once-holed tori conformally embedded into Y-0. It turns out that T-a[Y-0] and T-c[Y-0] have several properties in common. Our basic tool is a new notion, called a handle condition.