• Corpus ID: 204743793

# 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}
}
• Published 17 October 2019
• Mathematics
• arXiv: Geometric Topology
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

## Figures from this paper

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

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

### Discrete conformal geometry of polyhedral surfaces and its convergence

• Mathematics
Geometry &amp; Topology
• 2022
A BSTRACT . 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

### Combinatorial curvature flows for generalized circle packings on surfaces with boundary

• Mathematics
• 2022
In this paper, we investigate the deformation of generalized circle packings on ideally triangulated surfaces with boundary, which is the ( − 1 , − 1 , − 1) type generalized circle packing metric

### A new proof for global rigidity of vertex scaling on polyhedral surfaces

• Mathematics
• 2022
The vertex scaling for piecewise linear metrics on polyhedral surfaces was introduced by Luo [18], who proved the local rigidity by establishing a variational principle and conjectured the global

### The Weyl problem for unbounded convex domains in $\HH^3$

Let K ⊂ H be a convex subset in H with smooth, strictly convex boundary. The induced metric on ∂K then has curvature K > −1. It was proved by Alexandrov that if K is bounded, then it is uniquely

### Rigidity of Acute Triangulations of the Plane

. We show that a uniformly acute triangulation of the plane is rigid under Luo’s discrete conformal change, extending previous results on hexago- nal triangulations. Our result is a discrete analogue

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

• Xu Xu
• Mathematics
Calculus of Variations and Partial Differential Equations
• 2022
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

### Parameterized discrete uniformization theorems and curvature flows for polyhedral surfaces, II

• Mathematics
Transactions of the American Mathematical Society
• 2022
In this paper, we introduce a parameterized discrete curvature ( α -curvature) on polyhedral surfaces, which is a generalization of the classical discrete curvature. A discrete uniformization theorem

### Combinatorial Yamabe flow on hyperbolic bordered surfaces

• Mathematics
• 2022
This paper studies the combinatorial Yamabe ﬂow on hyperbolic bordered surfaces. We show that the ﬂow exists for all time and converges exponentially fast to conformal factor which produces a

### Hexagonal Geometric Triangulations

. 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

## References

SHOWING 1-10 OF 46 REFERENCES

### Polyhedral hyperbolic metrics on surfaces

Let S be a topologically finite surface, and g be a hyperbolic metric on S with a finite number of conical singularities of positive singular curvature, cusps and complete ends of infinite area. We

### On the classification of noncompact surfaces

3 in ?3.) In addition we prove the theorem, which so far as I know is new, that conversely, every nested triple of totally disconnected, compact, separable spaces occurs as the ideal boundary of some

### Fixed points, Koebe uniformization and circle packings

• Mathematics
• 1993
A domain in the Riemann sphere $$\hat{\mathbb{C}}$$ is called a circle domain if every connected component of its boundary is either a circle or a point. In 1908, P. Koebe [Ko1] posed the following

### Convergence of discrete conformal geometry and computation of uniformization maps

• Mathematics
Asian Journal of Mathematics
• 2019
The classical uniformization theorem of Poincaré and Koebe states that any simply connected surface with a Riemannian metric is conformally diffeomorphic to the Riemann sphere, or the complex plane

### A discrete uniformization theorem for polyhedral surfaces II

• Mathematics
Journal of Differential Geometry
• 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

### Transboundary extremal length

We introduce two basic notions, ‘transboundary extremal length’ and ‘fat sets’, and apply these concepts to the theory of conformal uniformization of multiply connected planar domains. A new short

### Intrinsic geometry of convex ideal polyhedra in hyperbolic 3-space

The main result is that every complete finite area hyperbolic metric on a sphere with punctures can be uniquely realized as the induced metric on the surface of a convex ideal polyhedron in

### Discrete conformal maps and ideal hyperbolic polyhedra

• Mathematics
• 2015
We establish a connection between two previously unrelated topics: a particular discrete version of conformal geometry for triangulated surfaces, and the geometry of ideal polyhedra in hyperbolic

### A discrete uniformization theorem for polyhedral surfaces

• Mathematics
Journal of Differential Geometry
• 2018
A discrete conformality for polyhedral metrics on surfaces is introduced in this paper which gen-eralizes earlier work on the subject. It is shown that each polyhedral metric on a surface is discrete

### Discrete Conformal Deformation: Algorithm and Experiments

• Jian Sun
• Mathematics, Computer Science
SIAM J. Imaging Sci.
• 2015
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.