Equivariant contactomorphisms of circular surfaces (Q2705848)

This very nice and well written note provides a short proof of a known characterization of circular CR structures. Originally, the first part is due to \textit{C. Epstein} [Invent. Math. 109, 351-403 (1992; Zbl 0786.32013)] and the second part to \textit{S. Semmes} [Mem. Am. Math. Soc. 98, No. 472 (1992; Zbl 0777.32015)]. The author shows that any equivariant CR structure on the unit sphere \(S^3\subset{\mathbf C}^2\) is embeddable as the boundary of a smooth strongly pseudoconvex circular domain \(D\) in \({\mathbf C}^2\). Conversely, any strongly pseudoconvex circular boundary \(\partial D\) in \({\mathbf C}^2\) is equivariantly contactomorphic to \(S^3\). An interesting trick is to write down the equation of \(D\) in the general form \(e^{u(\pi(z))} |z|^2\), where \(\pi:{\mathbf C}^2\to P_1({\mathbf C})\) is the standard projection and \(u:P_1({\mathbf C})\to {\mathbf R}\) is a smooth function. Quotienting by the circle action, an equivariant contactomorphism \(F:S^3\to \partial D\) gives rise to a symplectomorphism of \(P_1({\mathbf C})\) equipped with the two forms \(\omega_0\) and \(\omega_1\) obtained by quotienting the equivariant CR structures. By this observation, the characterization is reduced to the (well known) solvability of the \(\overline{\partial}\) over \(P_1({\mathbf C})\). Finally, based on an article of \textit{P. Tang} [Math. Z. 219, 49-69 (1995; Zbl 0823.32010)], the author shows that there exists a smooth strongly pseudoconvex circular domain \(D\subset {\mathbf C}^2\) such that no equivariant contactomorphism \(F: S^3\to \partial D\) minimizes the maximal distortion.