Uniformizing surfaces via discrete harmonic maps
Annales Henri Lebesgue, Volume 4 (2021), pp. 1767-1807.

Metadata

Keywords Discrete harmonic maps, finite weighted graphs, hyperbolic surfaces, Weil-Petersson geometry of Teichmüller spaces

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.


References

[CdV91a] Colin de Verdière, Yves Comment rendre géodésique une triangulation d’une surface ?, Enseign. Math., Volume 37 (1991) no. 3-4, pp. 201-212 | Zbl

[CdV91b] Colin de Verdière, Yves Un principe variationnel pour les empilements de cercles, Invent. Math., Volume 104 (1991) no. 3, pp. 655-669 | DOI | MR | Zbl

[EL81] Eells, James Jun.; Lemaire, Luc Deformations of metrics and associated harmonic maps, Proc. Indian Acad. Sci., Math. Sci., Volume 90 (1981) no. 1, pp. 33-45 | DOI | MR | Zbl

[Eps66] Epstein, David B. A. Curves von 2-manifolds and isotopies, Acta Math., Volume 115 (1966), pp. 83-107 | DOI | Zbl

[FM11] Farb, Benson; Margalit, Dan A Primer on Mapping Class Groups, Princeton Mathematical Series, 49, Princeton University Press, 2011 | DOI | Zbl

[FN03] Fenchel, Werner; Nielsen, Jakob Discontinuous Groups of Isometries in the Hyperbolic Plane, De Gruyter Studies in Mathematics, 29, Walter de Gruyter, 2003 | Zbl

[Hat02] Hatcher, Allen Algebraic Topology, Cambridge University Press, 2002 | Zbl

[Kee74] Keen, Linda Collars on Riemann surfaces, Discontinuous Groups and Riemann Surfaces (Proc. Conf., Univ. Maryland, College Park, Md., 1973) (Annals of Mathematics Studies), Volume 79, Princeton University Press (1974), pp. 263-268 | DOI | MR | Zbl

[Ker83] Kerckhoff, Steven P. The Nielsen realization problem, Ann. Math., Volume 117 (1983) no. 2, pp. 235-265 | DOI | MR | Zbl

[Koi79] Koiso, Nohirito Variation of harmonic mapping caused by a deformation of Riemannian metric, Hokkaido Math. J., Volume 8 (1979) no. 2, pp. 199-213 | MR | Zbl

[KS01] Kotani, Motoko; Sunada, Toshikazu Standard realizations of crystal lattices via harmonic maps, Trans. Am. Math. Soc., Volume 353 (2001) no. 1, pp. 1-20 | DOI | MR | Zbl

[KWZ18] Kim, Inkang; Wan, Xueyuan; Zhang, Genkai Plurisubharmonicity and geodesic convexity of energy function on Teichmüller space (2018) (https://arxiv.org/abs/1809.00255v1)

[Ste05] Stephenson, Kenneth Introduction to Circle Packing: The Theory of Discrete Analytic Functions, Cambridge University Press, 2005 | Zbl

[Sti92] Stillwell, John C. Geometry of Surfaces, Universitext, Springer, 1992 | DOI | Zbl

[Sun13] Sunada, Toshikazu Topological Crystallography. With a View Towards Discrete Geometric Analysis, Surveys and Tutorials in the Applied Mathematical Sciences, 6, Springer, 2013 | Zbl

[Thu78] Thurston, William P. The Geometry and Topology of Three-Manifolds (1978) (http://www.math.unl.edu/~mbrittenham2/classwk/990s08/public/thurston.notes.pdf/6a.pdf)

[Tro92] Tromba, Anthony J. Teichmüller Theory in Riemannian Geometry: based on lecture notes by Jochen Denzler, Lectures in Mathematics, ETH Zürich, Birkhäuser, 1992 | DOI | Zbl

[Wol87] Wolpert, Scott A. Geodesic length functions and the Nielsen problem, J. Differ. Geom., Volume 25 (1987) no. 2, pp. 275-296 | MR | Zbl

[Wol12] Wolf, Michael The Weil–Petersson Hessian of length of Teichmüller space, J. Differ. Geom., Volume 91 (2012) no. 1, pp. 129-169 | Zbl

[Yam99] Yamada, Sumio Weil–Petersson convexity of the energy functional on classical and universal Teichmüller spaces, J. Differ. Geom., Volume 51 (1999) no. 1, pp. 35-96 | MR | Zbl

[Yam14] Yamada, Sumio Local and global aspects of Weil–Petersson geometry, IRMA Lectures in Mathematics and Theoretical Physics, 19, European Mathematical Society (2014), pp. 43-111 | MR | Zbl

[Yam17] Yamada, Sumio On the Weil–Petersson convex geometry of Teichmüller space, Sugaku Expo., Volume 30 (2017) no. 2, pp. 159-186 | DOI | MR | Zbl