UNIFORMIZING SURFACES VIA DISCRETE HARMONIC MAPS

Toru Kajigaya; Ryokichi Tanaka

UNIFORMIZING SURFACES VIA DISCRETE HARMONIC MAPS

SCI - Mathematics

Research output: Contribution to journal › Article › peer-review

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 as a refinement of the Nielsen realization problem for the mapping class groups.

Original language	English
Journal	Unknown Journal
Publication status	Published - 2019 May 14

Keywords

Discrete harmonic maps
Finite weighted graphs
Hyperbolic surfaces
Weil-Petersson geometry of Teichmüller spaces

ASJC Scopus subject areas

General

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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 as a refinement of the Nielsen realization problem for the mapping class groups.",

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T1 - UNIFORMIZING SURFACES VIA DISCRETE HARMONIC MAPS

AU - Kajigaya, Toru

AU - Tanaka, Ryokichi

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Y1 - 2019/5/14

N2 - 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.

AB - 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.

KW - Discrete harmonic maps

KW - Finite weighted graphs

KW - Hyperbolic surfaces

KW - Weil-Petersson geometry of Teichmüller spaces

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