quasispheres

Quasispheres are a concept in geometric function theory and geometric topology. A quasisphere is a subset of three-dimensional Euclidean space that is the image of the standard two-dimensional sphere S^2 under a quasiconformal homeomorphism of R^3. Equivalently, a topological sphere S in R^3 is called a quasisphere if there exists a K-quasiconformal mapping f: R^3 → R^3 with f(S^2) = S for some distortion constant K ≥ 1. The case K = 1 corresponds to a Euclidean rigid motion of the sphere.

Quasiconformal maps generalize conformal maps by allowing bounded distortion of infinitesimal shapes. Roughly, a map is K-quasiconformal if it preserves shapes up to a controlled stretch everywhere, with the constant K bounding the eccentricity of infinitesimal ellipses.

Quasispheres inherit several geometric regularity properties: they are topological spheres; the images of smooth spheres under quasiconformal maps are typically not smooth but have controlled geometry. In particular, quasispheres are Ahlfors regular and linearly locally connected, and they often arise as boundaries of certain domains with quasiconformal structure; they can be highly non-smooth while remaining tame in the quasiconformal sense.

Not every topological sphere in R^3 is a quasisphere; wild spheres such as the Alexander horned sphere are not quasispheres. Quasispheres play a role in the study of Kleinian groups: the boundaries at infinity of quasi-Fuchsian groups are quasispheres.

History and references: The notion was developed in the theory of quasiconformal mappings by Gehring and Väisälä and has been used in three-dimensional geometric function theory and the study of 3-manifolds.