Uniformizing surfaces via discrete harmonic maps

Kajigaya, Toru; Tanaka, Ryokichi

Annales Henri Lebesgue, Tome 4 (2021), pp. 1767-1807.

Nous montrons que pour toute surface fermée de genre au moins deux et pour tout graphe pondéré fini remplissant la surface, il existe une métrique hyperbolique qui réalise le plus petit plongement harmonique en énergie de Dirichlet du graphe parmi une classe d’homotopie fixée et toutes les métriques hyperboliques sur la surface. Nous donnons des exemples explicites de telles surfaces hyperboliques par une nouvelle interprétation du problème de réalisation de Nielsen pour les groupes de difféotopies des surfaces.

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.

Classification : 32G15, 58E20, 05C10
Mots clés : Discrete harmonic maps, finite weighted graphs, hyperbolic surfaces, Weil-Petersson geometry of Teichmüller spaces

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