Two‐dimensional metric spheres from gluing hemispheres

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