Conformal Homeomorphisms of Strongly Pseudoconvex CR 3-Manifolds Are CR or Anti-CR: On a Question of Puqi Tang

Puqi Tang asked in the 1995 Several Complex Variables Problem List whether a homeomorphism between strongly pseudoconvex CR 3-manifolds that maps infinitesimal spheres of a Carnot–Carathéodory metric to infinitesimal spheres is CR; for this metric reading the answer is no as literally posed, because complex conjugation of the sphere \(S^3\)\({}\subset\mathbb{C}^2\) is such a map and is anti-CR, and yes under the convention of Korányi and Reimann that the map preserves the coorientation of the contact planes (\(f^*\theta'\)\({}=\lambda\theta\) with \(\lambda>0\)). More precisely, modulo Theorem 1.1 of Capogna, Citti, Le Donne and Ottazzi (J. Math. Pures Appl. 2019), a published theorem used without re-proof, we show the following. Let the Carnot–Carathéodory distances come from smooth Hermitian metrics on the contact planes, and let the metric distortion of the homeomorphism \(f\) be \(1\) at every point. Then \(f\) is a \(C^\infty\) diffeomorphism which on each connected component is CR or anti-CR, and it is CR exactly where it preserves the coorientation of the contact planes. Conversely, every such diffeomorphism has distortion \(1\) for every choice of Hermitian metrics. The manifolds need not be compact, connected or embeddable. The whole analytic content, the smoothness of \(1\)-quasiconformal maps between sub-Riemannian contact manifolds, is Theorem 1.1 of Capogna, Citti, Le Donne and Ottazzi; the answer is a corollary of that theorem and of elementary linear algebra, which for the flat model is already in the work of Cowling, Li, Ottazzi and Wu (2020); no new technique is claimed, and the novelty is low. Tang (1996) had proved the CR conclusion for locally embeddable manifolds and conformal homeomorphisms with locally integrable horizontal derivatives. The question is thus answered for the Carnot–Carathéodory metric reading, as a corollary of the 2019 theorem; the analytic reading of Tang (1996), in which conformality is defined through differentials that exist almost everywhere, remains open. This is an unrefereed note.