# Quasisymmetric Koebe Uniformization

@article{Merenkov2011QuasisymmetricKU, title={Quasisymmetric Koebe Uniformization}, author={Sergei Merenkov and Kevin Wildrick}, journal={arXiv: Metric Geometry}, year={2011} }

We study a quasisymmetric version of the classical Koebe uniformization theorem in the context of Ahlfors regular metric surfaces. In particular, we prove that an Ahlfors 2-regular metric surface X homeomorphic to a finitely connected domain in the standard 2-sphere is quasisymmetrically equivalent to a circle domain if and only if X is linearly locally connected and its completion is compact. We also give a counterexample in the countably connected case.

## 27 Citations

### Quasisymmetric Koebe uniformization with weak metric doubling measures

- MathematicsIllinois Journal of Mathematics
- 2021

We give a characterization of metric spaces quasisymmetrically equivalent to a finitely connected circle domain. This result generalizes the uniformization of Ahlfors 2-regular spaces by Merenkov and…

### Canonical parametrizations of metric surfaces of higher topology

- Mathematics
- 2022

We give an alternate proof to the following generalization of the uniformization theorem by Bonk and Kleiner. Any linearly locally connected and Ahlfors 2-regular closed metric surface is…

### Quasiconformal uniformization of metric surfaces of higher topology

- Mathematics
- 2022

A bstract . We establish the following uniformization result for metric spaces X of ﬁnite Hausdor ﬀ 2-measure. If X is homeomorphic to a smooth 2-manifold M with non-empty boundary, then we show that…

### Quantitative quasisymmetric uniformization of compact surfaces

- Mathematics
- 2016

Bonk and Kleiner showed that any metric sphere which is Ahlfors 2-regular and linearly locally contractible is quasisymmetrically equivalent to the standard sphere in a quantitative way. We extend…

### Quasisymmetric embeddings of slit Sierpi\'nski carpets

- Mathematics
- 2019

In this paper we study the problem of quasisymmetrically embedding metric carpets, i.e., spaces homeomorphic to the classical Sierpi\'nski carpet, into the plane. We provide a complete…

### Uniformization of two-dimensional metric surfaces

- MathematicsInventiones mathematicae
- 2016

We establish uniformization results for metric spaces that are homeomorphic to the Euclidean plane or sphere and have locally finite Hausdorff 2-measure. Applying the geometric definition of…

### Uniformization of two-dimensional metric surfaces

- Mathematics
- 2014

We establish uniformization results for metric spaces that are homeomorphic to the Euclidean plane or sphere and have locally finite Hausdorff 2-measure. Applying the geometric definition of…

### Quasispheres and metric doubling measures

- Mathematics
- 2017

Applying the Bonk-Kleiner characterization of Ahlfors 2-regular quasispheres, we show that a metric two-sphere $X$ is a quasisphere if and only if $X$ is linearly locally connected and carries a weak…

### Quasisymmetrically co-Hopfian Sierpi\'nski Spaces and Menger Curve

- Mathematics
- 2017

A metric space X is quasisymmetrically co-Hopfian if every quasisymmetric embedding of X into itself is surjective. We construct the first example of a metric space homeomorphic to the universal…

### Canonical parameterizations of metric disks

- MathematicsDuke Mathematical Journal
- 2020

We use the recently established existence and regularity of area and energy minimizing discs in metric spaces to obtain canonical parametrizations of metric surfaces. Our approach yields a new and…

## References

SHOWING 1-10 OF 28 REFERENCES

### QUASISYMMETRIC STRUCTURES ON SURFACES

- Mathematics
- 2007

We show that a locally Ahlfors 2-regular and locally linearly locally contractible metric surtace is locally quasisymmetrically equivalent to tne disk. We also discuss an application of this result…

### Quasisymmetric parametrizations of two-dimensional metric spheres

- Mathematics
- 2002

We study metric spaces homeomorphic to the 2-sphere, and find conditions under which they are quasisymmetrically homeomorphic to the standard 2-sphere. As an application of our main theorem we show…

### A Sierpiński carpet with the co-Hopfian property

- Mathematics
- 2010

Motivated by questions in geometric group theory we define a quasisymmetric co-Hopfian property for metric spaces and provide an example of a metric Sierpiński carpet with this property. As an…

### Quasisymmetric parametrizations of two‐dimensional metric planes

- Mathematics
- 2008

The classical uniformization theorem states that any simply connected Riemann surface is conformally equivalent to the disk, the plane, or the sphere, each equipped with a standard conformal…

### Quasiconformal maps in metric spaces with controlled geometry

- Mathematics
- 1998

This paper develops the foundations of the theory of quasiconformal maps in metric spaces that satisfy certain bounds on their mass and geometry. The principal message is that such a theory is both…

### Conformal dimension and Gromov hyperbolic groups with 2-sphere boundary

- Mathematics
- 2005

Suppose G is a Gromov hyperbolic group, and @1G is quasisymmetrically homeomorphic to an Ahlfors Q–regular metric 2–sphere Z with Ahlfors regular conformal dimension Q. Then G acts discretely,…

### Quasiconformal geometry of fractals

- Mathematics
- 2006

Many questions in analysis and geometry lead to problems of quasiconformal geometry
on non-smooth or fractal spaces. For example, there is a close relation of this subject
to the problem of…

### Quasicircles and Bounded Turning Circles Modulo bi-Lipschitz Maps

- Mathematics
- 2010

We construct a catalog, of snowflake type metric circles, that describes all metric quasicircles up to \bl\ equivalence. This is a metric space analog of a result due to Rohde. Our construction also…

### The quasiconformal Jacobian problem

- Mathematics
- 2006

Which nonnegative functions can arise, up to a bounded multiplicative error, as Jacobian determinants Jf(x) = det(Df(x)) of quasiconformal mappings f :R n → R, n ≥ 2? Which metric spaces are…

### 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…