Uniformizing surfaces via discrete harmonic maps
Annales Henri Lebesgue, Volume 4 (2021), pp. 1767-1807.


Keywords Discrete harmonic maps, finite weighted graphs, hyperbolic surfaces, Weil-Petersson geometry of Teichmüller spaces


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 through a new interpretation of the Nielsen realization problem for the mapping class groups.


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