# A discrete uniformization theorem for polyhedral surfaces II

@article{Gu2018ADU, title={A discrete uniformization theorem for polyhedral surfaces II}, author={Xianfeng Gu and Ren Guo and Feng Luo and Jian Sun and Tianqi Wu}, journal={Journal of Differential Geometry}, year={2018} }

A discrete conformality for hyperbolic polyhedral surfaces is introduced in this paper. This discrete conformality is shown to be computable. It is proved that each hyperbolic polyhedral metric on a closed surface is discrete conformal to a unique hyperbolic polyhedral metric with a given discrete curvature satisfying Gauss-Bonnet formula. Furthermore, the hyperbolic polyhedral metric with given curvature can be obtained using a discrete Yamabe flow with surgery. In particular, each hyperbolic…

## 76 Citations

A discrete uniformization theorem for polyhedral surfaces

- MathematicsJournal of Differential Geometry
- 2018

A discrete conformality for polyhedral metrics on surfaces is introduced in this paper which generalizes earlier work on the subject. It is shown that each polyhedral metric on a surface is discrete…

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Our recent joint work with D. Gu established a discrete version of the uniformization theorem for compact polyhedral surfaces. In this talk, we prove that discrete uniformizaton maps converge to…

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The paper proves a result on the convergence of discrete conformal maps to the Riemann mappings for Jordan domains. It is a counterpart of Rodin-Sullivan’s theorem on convergence of circle packing…

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We provide a constructive, variational proof of Rivin's realization theorem for ideal hyperbolic polyhedra with prescribed intrinsic metric, which is equivalent to a discrete uniformization theorem…

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We provide a constructive, variational proof of Rivin’s realization theorem for ideal hyperbolic polyhedra with prescribed intrinsic metric, which is equivalent to a discrete uniformization theorem…

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A definition of discrete conformality for triangulated surfaces with flat cone metrics and an algorithm for solving the problem of prescribing curvature to deform the metric discrete conformally so that the curvature of the resulting metric coincides with the prescribed curvature.

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We introduce a new type of discrete conformal structures on surfaces with boundary, which have nice interpolations in 3-dimensional hyperbolic geometry. Then we prove the global rigidity of the new…

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This paper describes a numerical method for surface parameterization, yielding maps that are locally injective and discretely conformal in an exact sense. Unlike previous methods for discrete…

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