Hyperbolic Cauchy-Riemann (CR) singularities and KAM-like theory for holomorphic involutions

We consider the real analytic perturbed Bishop quadric surfaces in (C^2,0), having an isolated CR singularity at the origin. There are two kinds of stable CR singularities: elliptic and hyperbolic. The elliptic case was studied by Moser-Webster, where they showed that such a surface is locally, near the CR singularity, holomorphically equivalent to normal form from which lots of geometric features can be read off.

We focus on the hyperbolic case. As shown by Moser-Webster, such a surface can be transformed to a formal normal form by a formal change of coordinates (but usually not holomorphic in any neighborhood of origin). For a non-degenerate real analytic surface M having a hyperbolic CR singularity at the origin, we prove the existence of Whitney smooth family of curves intersecting M along holomorphic hyperbolas. This is shown by a KAM-like theorem for a pair of holomorphic involutions at the origin, a common fixed point. Joint work with L. Stolovitch.