Harmonic maps between ideal 2-dimensional simplicial complexes
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Harmonic maps between ideal 2-dimensional simplicial complexes Brian Freidin1
· Victòria Gras Andreu2
Received: 3 November 2018 / Accepted: 4 February 2020 © Springer Nature B.V. 2020
Abstract We prove existence and regularity results for energy minimizing maps between ideal hyperbolic 2-dimensional simplicial complexes. The spaces in question were introduced by Charitos–Papadopoulos, who describe their Teichmüller spaces and some compactifications. This work is a first step in introducing harmonic map theory into the Teichmüller theory of these spaces. Keywords Harmonic maps · Simplicial complexes · Hyperbolic geometry · Teichmuller theory Mathematics Subject Classification 53C43
1 Introduction Charitos and Papadopoulos [1] study finite 2-dimensional simplicial complexes. They describe how to endow each face (with vertices removed) with the structure of an ideal hyperbolic triangle, and special parameters describing the ways to glue the faces together to form the complex, characterizing those metrics that are complete. They compute the dimension of the Teichmüller space of complete ideal hyperbolic metrics on a complex, and describe a compactification of this space in terms of special measured foliations. The theory of harmonic maps has been applied fruitfully in Teichmüller theory in many ways. To a harmonic map f between surfaces, one associates its Hopf differential φ(z)dz 2 , the (2, 0) part of the pull-back of the target metric by f , and the harmonicity of f is equivalent to the holomorphicity of the Hopf differential. Sampson [20] and Wolf [23] show that for a fixed metric g0 on a surface S of genus γ , the map that associates to a hyperbolic metric g on
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Brian Freidin [email protected] Victòria Gras Andreu [email protected]
1
Pacific Institute for the Mathematical Sciences, University of British Columbia, Vancouver, BC, Canada
2
Brown University, Providence, USA
123
Geometriae Dedicata
S the Hopf differential of the harmonic map f : (S, g0 ) → (S, g) is a homeomorphism from the Teichmüller space of S to the space of holomorphic quadratic differentials on (S, g0 ). In another direction, Gerstenhaber and Rauch propose in [9] a variational characterization of Teichmüller mappings, those that minimize the complex dilatation in their isotopy class, via harmonic maps. In [16], in fact, Kuwert shows that the Teichmüller map is harmonic with respect to a particular singular flat metric in the conformal class of the target. In [17] Mese resolves the Gerstenhaber-Rauch principle by producing the Teichmüller map by variational methods. The aim of this paper is to introduce harmonic map theory into the Teichmüller theory of the ideal hyperbolic complexes of [1], with an eye towards emulating the arguments of [17], by first establishing the existence of harmonic maps. Existence results for harmonic maps goes back at least as far as the work of Eells and Sampson [7] from ’64, where they prove existence under curvature and completeness hypotheses by the heat flow method. The study of harmonic maps
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