The Teichmüller distance on the space of spherical CR structures
It is well known that quasiconformal mappings between Riemann surfaces can be used to define a distance, the Teichmüller distance, on the Teichmüller space of Riemann surfaces with a fixed genus. Such a construction was generalized by H.~Izeki to define a distance on the space of conformally flat structures. The complex counterpart of a conformally flat manifold is the spherical CR manifold. Denote by \(\text{Diff\,}_0^r(M)\) the group of \(C^n\) diffeomorphisms of the manifold \(M\) isotropic to the identity mapping. Define the space \[ \mathcal T(M) = \{\text{spherical CR structures on } M\}/\text{Diff\,}_0^\infty(M). \] Denote by \([J]\) the element of \(\mathcal T(M)\) represented by a spherical CR structure \(J\). The topology of the space \(\mathcal T(M)\) is simply given by that of the space \(\mathcal T^\infty(M)\) constructed from developing maps. This is the counterpart of the Teichmüller space in the spherical CR case. There is a natural definition of quasiconformal mappings between spherical CR manifolds. It is shown that the distortion of a quasiconformal mapping \(f\) between two spherical CR manifolds \((M, J)\) and \((N, J')\), denoted by \(K(f;(M, J),(N,J'))\), is well defined. The Teichmüller pseudodistance \(d\) on the space \(\mathcal T(M)\) defined by \[ d([J], [J'])=\inf_f\log K (f, (M, J), (M, J)'),\tag{1} \] for \([J], [J']\in\mathcal T(M)\), is well defined, i.e. the right-hand side of (1) is independent of the choices of spherical CR structures \(J\in [J]\) and \(J'\in [J']\), where the infimum is taken over all orientation preserving quasiconformal mappings \(f\) from \((M, J)\) to \((M, J')\), which are homotopic to the identity as homeomorphisms. Let \(M\) be a \(C^\infty\) orientable, connected, compact manifold without boundary. Then the Teichmüller pseudodistance on the space \(\mathcal T(M)\) is shown to be a complete distance by establishing a compactness result. If a spherical CR structure \(J'\) is a smooth deformation of the other spherical CR structure \(J\), then \(d([J], [J'])\) is finite.
quasiconformal mappings between spherical CR manifolds
scientific article; zbMATH DE number 5127958