Abstract:
This joint work with Z. Zhao (Nice) is concerned with the
geometry of germs of real analytic surfaces in $(\mathbb{C}^2,0)$ having
an isolated Cauchy–Riemann (CR) singularity at the origin. These are
perturbations of Bishop quadrics. There are two kinds of CR
singularities stable under perturbation : elliptic and hyperbolic.
Elliptic case was studied by Moser–Webster
who 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.
In this talk we focus on perturbations of hyperbolic quadrics.
As was shown by Moser–Webster, such a surface can be transformed to a
formal normal form by a formal change of coordinates that may not
be holomorphic in any neighborhood of the origin.
Given a non-degenerate real analytic surface $M$ in
$(\mathbb{C}^2,0)$ having a hyperbolic CR singularity at the
origin, we prove the existence of
Whitney smooth family of holomorphic curves intersecting $M$ along
holomorphic hyperbolas. This is the very first result concerning hyperbolic CR
singularity not equivalent to quadrics.
This is a consequence of a non-standard KAM-like theorem for pair of
germs of holomorphic involutions $\{\tau_1,\tau_2\}$ at the origin, a
common fixed point. We show that such a pair has large amount of
invariant analytic sets biholomorphic to $\{z_1z_2=const\}$ (which is
not a torus) in a neighborhood of the origin, and that they are
conjugate to restrictions of linear maps on such invariant sets.