We show that for any closed surface of genus greater than one and for any finite weighted graph filling the surface, there exists a hyperbolic metric which realizes the least Dirichlet energy harmonic embedding of the graph among a fixed homotopy class and all hyperbolic metrics on the surface. We give explicit examples of such hyperbolic surfaces as a refinement of the Nielsen realization problem for the mapping class groups.
|Publication status||Published - 2019 May 14|
- Discrete harmonic maps
- Finite weighted graphs
- Hyperbolic surfaces
- Weil-Petersson geometry of Teichmüller spaces
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