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Harmonic maps between ideal 2-dimensional simplicial complexes. (English) Zbl 1453.53065

C. Charitos and A. Papadopoulos [Glasg. Math. J. 43, No. 1, 39–66 (2001; Zbl 0977.57003)] developed a theory of hyperbolic structures on a finite two-dimensional simplicial complex \(X\). An ideal hyperbolic structure on \(X\) is a complete metric on the subspace \(X\backslash S\), where \(S\) is the set of vertices of \(X\), such that the induced metric on each face of \(X\) is isometric to an ideal triangle in hyperbolic 2-spaces. Such a structure is not unique, because it can be changed by varying the gluing maps between edges of each triangle. The moduli space of such structures is called the Teichmüller space \(\mathcal{T}(X)\) of \(X\).
The authors develop the theory of harmonic maps between such ideal hyperbolic structures, with the eventual goal of using variational methods to find Teichmüller maps, which are those maps that minimize the complex dilation in their isotopy class. Such an approach for finding Teichmüller maps was first proposed by M. Gerstenhaber and H. E. Rauch [Proc. Nat. Acad. Sci. USA 40, 808–812, 991–994 (1954; Zbl 0056.07502)] for Riemann surfaces and was subsequently used in various settings by C. Mese [Calc. Var. Partial Differ. Equ. 21, No. 1, 15–46 (2004; Zbl 1065.30019)] and E. Kuwert [Math. Z. 221, No. 3, 421–436 (1996; Zbl 0871.58028)].
Given two ideal hyperbolic structures, \(\sigma\) and \(\tau\), on a simplicial complex \(X\) as above, \(W^{1,2}\) denotes the space of all maps \(f\) from \((X\backslash S,\sigma)\) to \((X\backslash S,\tau)\) mapping each vertex, edge and face to itself and having finite energy \(E(f)= \sum_T \int_T |\nabla f|^2\), where the sum is taken over all faces \(T\) in \(X\). The class \(\mathcal{D}\subset W^{1,2}\) denotes the class of maps \(f\) in \(W^{1,2}\) such that \(f|T\) is a diffeomorphism for each face \(T\) of \(X\).
The main results proved by the authors are:
(i)
There exists a map \(H\in\mathcal{D}\) of finite energy.
(ii)
There exist energy minimizing maps \(u\in W^{1,2}\) and \(u_{\mathcal{D}}\in \overline{\mathcal{D}}\), the weak closure of \(\mathcal{D}\) in \(W^{1,2}\). Moreover, both \(u\) and \(u_{\mathcal{D}}\) are homotopic to the map \(H\) constructed in (i).
(iii)
For any face \(T\) of \(X\), the restrictions \(u|T\) and \(u_{\mathcal{D}}|T\) are analytic harmonic maps in the interior of \(T\), and \(u|T\) is analytic up to the boundary of \(T\).

MSC:

53C43 Differential geometric aspects of harmonic maps
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
57M50 General geometric structures on low-dimensional manifolds
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References:

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