Two-dimensional metric spheres from gluing hemispheres

TL;DR

This paper investigates the geometric and quasiconformal properties of metric spheres formed by gluing hemispheres of a Euclidean sphere, establishing conditions for their equivalence to the standard sphere.

Contribution

It characterizes when such glued spheres are quasiconformally equivalent to the Euclidean sphere based on properties of the gluing homeomorphism g.

Findings

g is a welding homeomorphism with removable curves if Z is quasiconformal to the sphere

g is bi-Lipschitz iff Z admits a 1-quasiconformal parametrization with comparable Jacobian

If g's inverse is absolutely continuous and extends with exponentially integrable distortion, Z is quasiconformally equivalent to the sphere

Abstract

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.