Twodimensional metric spheres from gluing hemispheres
Abstract
We study metric spheres Z obtained by gluing two hemispheres of the Euclidean sphere along an orientationpreserving homeomorphism mapping the equator onto itself, where the distance on Z is the canonical distance that is locally isometric to the spherical distance off the seam. We show that if Z is quasiconformally equivalent to the sphere, in the geometric sense, then g is a welding homeomorphism with conformally removable welding curves. We also show that g is biLipschitz if and only if Z has a 1quasiconformal parametrization whose Jacobian is comparable to the Jacobian of a quasiconformal mapping from the Euclidean sphere onto itself. Furthermore, we show that if the inverse of g is absolutely continuous and g admits a homeomorphic extension with exponentially integrable distortion, then Z is quasiconformally equivalent to the Euclidean sphere.
 Publication:

arXiv eprints
 Pub Date:
 June 2021
 arXiv:
 arXiv:2106.01295
 Bibcode:
 2021arXiv210601295I
 Keywords:

 Mathematics  Complex Variables;
 Mathematics  Metric Geometry;
 Primary 30L10;
 Secondary 30C65;
 28A75;
 51F99;
 52A38
 EPrint:
 30 pages