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A once-holed torus is a noncompact Riemann surface of genus one with exactly one boundary component.We address the existence problem of handle-preserving holomorphic mappings of once-holed tori into a given Riemann surface of positive genus.The Teichmueller space of a once-holed torus is a 3-dimensional real analytic manifold with boundary,where the local coordinate systems are given in terms of extremal lengths.We investigate geometric properties of the set of once-holed tori allowing such holomorphic mappings.