Two-dimensional metric spheres from gluing hemispheres

We study metric spheres Z obtained by gluing two hemispheres of the Euclidean sphere along an orientation-preserving 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...

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Bibliographic Details
Published in:arXiv.org 2021-06
Main Author: Ikonen, Toni
Format: Article
Language:English
Subjects:
Equivalence
Euclidean geometry
Gluing
Hemispheres
Mapping
Parameterization
Topology
Welding
Online Access:Get full text
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Summary:We study metric spheres Z obtained by gluing two hemispheres of the Euclidean sphere along an orientation-preserving 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 bi-Lipschitz if and only if Z has a 1-quasiconformal 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.
ISSN:2331-8422