Abstract
The aim of this paper is to study finite convex polyhedra in three dimensional hyperbolic space \({\mathbb {H}}^3\). We characterize the quasiconformal deformation space of each finite convex polyhedron. As a corollary, we obtain some results on finite circle patterns in the Riemann sphere with dihedral angle \(0\le \Theta < \pi \). That is, for any circle pattern on \(\hat{\mathbb {C}}\), its quasiconformal deformation space can be naturally identified with the product of the Teichmüller spaces of its interstices.
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Notes
A closed local geodesic is the image of a standard circle under a locally distance minimizing mapping. At any interior cone point, it subtends an angle of at least \(\pi \) on either side. When a geodesic passes through a boundary cone point, it subtends an angle at least \(\pi \) on the side of the region Q.
A lung is a piece of a sphere bounded by two geodesic arcs.
The strong maximal principle states that: if a non-constant bounded harmonic function attains its minimal (resp. maximal) at \(z_0 \in \partial \Omega \), and the boundary \(\partial \Omega \) satisfies an interior sphere condition at \(z_0\), then the outer normal derivative of u at \(z_0\), if it exists, satisfies the strict inequality \({\frac{\partial u}{\partial n}} (z_0)>0\) (resp. \(<0\)). See e.g. Lemma 3.4 of [3].
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Acknowledgments
We wish to thank Professor Zhengxu He for suggesting this project. We are also very grateful to the anonymous referee for a careful reading of the manuscript and for many helpful suggestions.
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Both of authors were partially supported by NSFC of China (No. 11471318).
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Huang, X., Liu, J. Characterizations of circle patterns and finite convex polyhedra in hyperbolic 3-space. Math. Ann. 368, 213–231 (2017). https://doi.org/10.1007/s00208-016-1433-y
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DOI: https://doi.org/10.1007/s00208-016-1433-y