In this thesis, we construct polynomial growth harmonic maps from once-punctured Riemann surfaces of any finite genus to any even-sided, regular, ideal polygon in the hyperbolic plane. We also establish their uniqueness within a class of maps which differ by exponentially decaying variations.

Previously, harmonic maps from C (which are conformally once-punctured spheres) to H^2 have been parameterized by holomorphic quadratic differentials on C. Our harmonic maps, mapping a genus g>1 punctured surface to a k-sided polygon, correspond to meromorphic quadratic differentials with one pole of order (k+2) at the puncture and (4g+k−2) zeros (counting multiplicity). In this way, we can associate to these maps a holomorphic quadratic differential on the punctured Riemann surface domain.

As an example, we explore a special case of our theorems: the unique harmonic map from a punctured square torus to an ideal square. We use the symmetries of the map to deduce the three possibilities for its Hopf differential.