Wolf, Michael2017-08-022017-08-022016-052016-04-20May 2016Huang, Andy C. "Handle crushing harmonic maps between surfaces." (2016) Diss., Rice University. <a href="https://hdl.handle.net/1911/96251">https://hdl.handle.net/1911/96251</a>.https://hdl.handle.net/1911/96251In 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.application/pdfengCopyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.Harmonic mapsDifferential geometryThesis2017-08-02