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arXiv · 1905.05427

Uniformizing surfaces via discrete harmonic maps

Toru Kajigaya·Ryokichi Tanaka
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Abstract

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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Toru Kajigaya, Ryokichi Tanaka. 2019-05-14. Uniformizing surfaces via discrete harmonic maps. https://arxiv.org/abs/1905.05427

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