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. 

Figures from this paper

Quasisymmetric Koebe uniformization with weak metric doubling measures

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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