Uniformizing surfaces via discrete harmonic maps
Differential Geometry
2020-07-27 v2 Metric Geometry
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.
Cite
@article{arxiv.1905.05427,
title = {Uniformizing surfaces via discrete harmonic maps},
author = {Toru Kajigaya and Ryokichi Tanaka},
journal= {arXiv preprint arXiv:1905.05427},
year = {2020}
}
Comments
31 pages, 5 figures