# Koebe conjecture and the Weyl problem for convex surfaces in hyperbolic 3-space

@article{Luo2019KoebeCA, title={Koebe conjecture and the Weyl problem for convex surfaces in hyperbolic 3-space}, author={Feng Luo and Tianqi Wu}, journal={arXiv: Geometric Topology}, year={2019} }

We prove that the Koebe circle domain conjecture is equivalent to the Weyl type problem that every complete hyperbolic surface of genus zero is isometric to the boundary of the hyperbolic convex hull of the complement of a circle domain. It provides a new way to approach the Koebe's conjecture using convex geometry. Combining our result with the work of He-Schramm on the Koebe conjecture, one establishes that every simply connected non-compact polyhedral surface is discrete conformal to the…

## 16 Citations

### On the Weyl problem for complete surfaces in the hyperbolic and anti-de Sitter spaces

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The classical Weyl problem (solved by Lewy, Alexandrov, Pogorelov, and others) asks whether any metric of curvature K ≥ 0 on the sphere is induced on the boundary of a unique convex body in R3. The…

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### A new class of discrete conformal structures on surfaces with boundary

- MathematicsCalculus of Variations and Partial Differential Equations
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We introduce a new class 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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- MathematicsTransactions of the American Mathematical Society
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. It is well-known that the Euclidean plane has a standard 6-regular geodesic triangulation , and the unit sphere has a 5-regular geodesic triangulation, which is induced from the regular…

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