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On the Teichmüller theory of circle patterns. (English) Zbl 1318.30017

While already classical, circle patterns and their applications have become an important tool of modern mathematics after being used by Thurston in his approach to Andreev’s Theorem [W. Thurston, “The geometry and topology of \(3\)-manifolds”, preliminary version, Chap. 1–9, Princeton Univ. (1990)]. Their importance was reinforced more recently by B. Chow and F. Luo’s work [J. Differ. Geom. 63, No. 1, 97–129 (2003; Zbl 1070.53040)] on the discrete Ricci flow and its various practical applications; see, e.g., [X. D. Gu (ed.) and S.-T. Yau (ed.), Computational conformal geometry. With CD-ROM. Somerville, MA: International Press; Beijing: Higher Education Press (2008; Zbl 1144.65008)].
Given a graph \(G = (V,E)\), its realization as a circle pattern \(P\) in \(\widehat{\mathbb{C}}\) determines interstices \(I\), i.e., components of \(\widehat{\mathbb{C}} \setminus \bigcup_{v \in V}D(v)\), together with a marking of the circular arcs or circles on their respective boundaries, where \(D(v)\) denotes the open disk bounded by the circle \(P(v)\) corresponding to the vertex \(v \in V\).
The Teichmüller space \(\mathcal{T}_I\) of \(I\) is then defined as the space of all equivalence classes of quasiconformal mappings \(h:I \rightarrow \widehat{\mathbb{C}}\).
The main result of the paper is to show that given a graph (apart from the 1-skeleton of a tetrahedron), embedded in \(\widehat{\mathbb{C}}\), with intersection (dihedral) angles \(\Theta \in [0,\pi/2]\), and satisfying a number of additional technical conditions, the space of equivalence classes of circle packings \(P\) realizing \((G,\Theta)\) admits a natural identification with the product of Teichmüller spaces \(\prod_1^p\mathcal{T}_{I_i}\), where \(\{I_1,I_2,\dots,I_p\}\) denote the interstices of \(P\).
The authors defer for later study the more difficult case \(\Theta_P \in [0,\pi)\) (together with some related problems).

MSC:

30C35 General theory of conformal mappings
30F60 Teichmüller theory for Riemann surfaces
30C80 Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination
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