# Two-dimensional metric spheres from gluing hemispheres

@inproceedings{Ikonen2021TwodimensionalMS, title={Two-dimensional metric spheres from gluing hemispheres}, author={Toni Ikonen}, year={2021} }

We study metric spheres (Z, dZ) obtained by gluing two hemispheres of S2 along an orientation-preserving homeomorphism g : S1 → S1, where dZ is the canonical distance that is locally isometric to S2 off the seam. We show that if (Z, dZ) is quasiconformally equivalent to S 2, in the geometric sense, then g is a welding homeomorphism with conformally removable welding curves. We also show that g is bi-Lipschitz if and only if (Z, dZ) has a 1quasiconformal parametrization whose Jacobian is…

## One Citation

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## References

SHOWING 1-10 OF 60 REFERENCES

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…

Geometric and analytic quasiconformality in metric measure spaces

- Mathematics
- 2010

We prove the equivalence between geometric and analytic definitions of quasiconformality for a homeomorphism $f\colon X\rightarrow Y$ between arbitrary locally finite separable metric measure spaces,…

Harmonic measure,L2-estimates and the Schwarzian derivative

- Mathematics
- 1994

We consider several results, each of which uses some type of “L2” estimate to provide information about harmonic measure on planar domains. The first gives an a.e. characterization of tangent points…

Geometric characterizations of p-Poincaré inequalities in the metric setting

- Mathematics
- 2016

We prove that a locally complete metric space endowed with a doubling measure satisfies an ∞-Poincar´e inequality if and only if given a null set, every two points can be joined by a quasiconvex…

Extension of boundary homeomorphisms to mappings of finite distortion

- Mathematics
- 2021

We provide sufficient conditions so that a homeomorphism of the real line or of the circle admits an extension to a mapping of finite distortion in the upper half-plane or the disk, respectively.…

Uniformization of metric surfaces using isothermal coordinates

- MathematicsAnnales Fennici Mathematici
- 2021

We establish a uniformization result for metric surfaces – metric spaces that are topological surfaces with locally finite Hausdorff 2-measure. Using the geometric definition of quasiconformality,…

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…

ON BOUNDARY HOMEOMORPHISMS OF TRANS-QUASICONFORMAL MAPS OF THE DISK

- Mathematics
- 2008

This paper studies boundary homeomorphisms of trans-quasiconformal maps of the unit disk. Motivated by Beurling-Ahlfors's well-known quasisymme- try condition, we introduce the \scalewise" and…

Complex Variables and Elliptic Equations

- 2018

Newton flows are dynamical systems generated by a continuous, desingularized Newton method for mappings from a Euclidean space to itself. We focus on the special case of meromorphic functions on the…

Maximal metric surfaces and the Sobolev-to-Lipschitz property

- Mathematics
- 2019

We find maximal representatives within equivalence classes of metric spheres. For Ahlfors regular spheres these are uniquely characterized by satisfying the seemingly unrelated notions of…