Geometry of hyperbolic Cauchy-Riemann singularities and KAM-like theory for holomorphic involutions

TL;DR

This paper investigates the geometry of hyperbolic CR singularities in real analytic surfaces in complex 2-space, establishing a KAM-like theory for holomorphic involutions and demonstrating the existence of invariant holomorphic curves intersecting the surface.

Contribution

It introduces a novel KAM-like theorem for pairs of holomorphic involutions and proves the existence of invariant analytic sets near hyperbolic CR singularities, extending understanding beyond quadrics.

Findings

Existence of a Whitney smooth family of holomorphic curves intersecting the surface.

Invariant analytic sets biholomorphic to z_1z_2=const are abundant near the singularity.

These invariant sets are conjugate to linear maps, revealing geometric structure.

Abstract

This article 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 {\it Bishop quadrics}. There are two kinds of CR singularities stable under perturbation~: {\it elliptic} and {\it hyperbolic}. Elliptic case was studied by Moser-Webster \cite{moser-webster} who showed that such a surface is locally, near the CR singularity, holomorphically equivalent to {\it normal form} from which lots of geometric features can be read off. In this article we focus on perturbations of {\it hyperbolic} quadrics. As was shown by Moser-Webster \cite{moser-webster}, such a surface can be transformed to a formal {\it normal form} by a formal change of coordinates that may not be holomorphic in any neighborhood of the origin. Given a {\it non-degenerate} real analytic surface $M$ in $(\mathbb{C}^2,0)$ having a {\it hyperbolic} CR singularity at the origin, we prove the existence of a non-constant Whitney smooth family of connected 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.