Two‐dimensional metric spheres from gluing hemispheres

Abstract

We study metric spheres ( Z , d Z ) $(Z, d_{Z} )$ obtained by gluing two hemispheres of S 2 $\mathbb {S}^{2}$ along an orientation-preserving homeomorphism g : S 1 → S 1 $g \colon \mathbb {S}^{1} \rightarrow \mathbb {S}^{1}$ , where d Z $d_{Z}$ is the canonical distance that is locally isometric to S 2 $\mathbb {S}^{2}$ off the seam. We show that if ( Z , d Z ) $(Z, d_{Z} )$ is quasiconformally equivalent to S 2 $\mathbb {S}^{2}$ , in the geometric sense, then g $g$ is a welding homeomorphism with conformally removable welding curves. We also show that g $g$ is bi-Lipschitz if and only if ( Z , d Z ) $(Z, d_{Z} )$ has a 1-quasiconformal parametrization whose Jacobian is comparable to the Jacobian of a quasiconformal mapping h : S 2 → S 2 $h \colon \mathbb {S}^{2} \rightarrow \mathbb {S}^{2}$ . Furthermore, we show that if g − 1 $g^{-1}$ is absolutely continuous and g $g$ admits a homeomorphic extension with exponentially integrable distortion, then ( Z , d Z ) $(Z, d_{Z} )$ is quasiconformally equivalent to S 2 $\mathbb {S}^{2}$ .