论文标题
双曲线牛chy-riemann奇异性和类似KAM的理论的几何形状
Geometry of hyperbolic Cauchy-Riemann singularities and KAM-like theory for holomorphic involutions
论文作者
论文摘要
本文涉及$（\ Mathbb {C}^2,0）$中的真实分析表面细菌的几何形状，其起源是孤立的cauchy-riemann（cr）奇异性。这些是{\ it Bishop Quadrics}的扰动。在扰动〜：{\ it椭圆形}和{\ it双曲线}下，有两种Cr奇点稳定。 Moser-Webster \ Cite {Moser-Webster}研究了椭圆案例，他表明这种表面在局部，靠近Cr奇异性，与{\ it正常形式}相当于{\ it正常形式}，从中可以读取许多几何特征。 在本文中，我们着重于{\ it双曲线} Quadrics的扰动。正如Moser-Webster \ cite {Moser-webster}所示的那样，可以通过正式变化的坐标变化来将这种表面转化为正式的{\ it正常形式}，而坐标的正式变化可能不会在原点的任何邻里中塑形。 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.这是关于双曲线cr奇异性的第一个结果，不等于四边形。 这是一个非标准的KAM样定理的结果，用于holomorphic wentureds $ \ {τ_1，τ_2\} $在原点（一个共同的固定点）。我们表明，这样的一对具有大量不变的分析集，将Biholomormormormormorphic到$ \ {Z_1Z_2 = CONST \} $（这不是原始附近的圆环），并且它们共轭以限制了这种不变集的线性图。
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 $\{τ_1,τ_2\}$ 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.
