Two‐dimensional metric spheres from gluing hemispheres

Abstract

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 S2, 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 1-quasiconformal parametrization whose Jacobian is comparable to the Jacobian of a quasiconformal mapping h:S2→S2. Furthermore, we show that if g−1 is absolutely continuous and g admits a homeomorphic extension with exponentially integrable distortion, then (Z,dZ) is quasiconformally equivalent to S2.