Wei Wang: Quasiconformal mappings, quasiCR mappings and Beltrami-type systems on some CR manifolds

A hypersurface in the complex Euclidean space is a CR
manifold: the complex structure is only defined on its horizontal subspace. When it is strongly pseudoconvex, there is a naturally induced Carnot-Caratheodory distance over it. We show that quasiconformal mappings with respect to this distance satisfy a Beltrami-type system, from which we see that 1-quasiconformal mappings are CR and are linearizable in some cases (Liouville-type Theorem). There also exist conformal mappings with respect to indefinite metrics (e.g., Lorentz metric), but it is difficult to define "quasiconformal" deformation of such conformal mappings. However, in the complex case, we can define quasiCR mappings over any quadratic CR manifolds by using Beltrami-type systems directly, and show their Holder equicontinuity and stability when the manifolds are strongly 2-pseudoconcave. The later condition is connected to the subellipticity of the Beltrami-type system.