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.Comment: 31 pages, 5 figure