  In this paper, 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 once-punctured spheres to the hyperbolic plane
have been parameterized by holomorphic quadratic differentials on the complex
plane. Our harmonic maps, mapping a genus g>1 domain to a k-sided polygon,
correspond to meromorphic quadratic differentials having one pole of order
(k+2) 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 specialize our theorems to obtain a harmonic
map from a punctured square torus to an ideal square, and deduce the five
possibilities for the divisor of its Hopf differential.
