Asymptotic curves on real analytic surfaces in \({\mathbb{C}}^ 2\)
The author considers a real two-dimensional analytic surface \(M\subset {\mathbb{C}}^ 2\) in the neighborhood of an isolated point p with a complex tangent. Let \(p=0=origin\) of \({\mathbb{C}}^ 2\). The author is interested in local properties of M, invariant under holomorphic local coordinate transformations. If M is given in a neighborhood of 0 by \(z_ 2 = az^ 2_ 1 + bz_ 1\bar z_ 1 + c\bar z_ 1 +...,\) \(b\neq 0\), then a holomorphic coordinate change leads to \(b=1\), \(a=c=\gamma \geq 0\) where \(\gamma\) is a biholomorphic invariant first considered by Bishop. In the hyperbolic case \(\gamma >\), if a solution \(\lambda\in {\mathbb{C}}\) of \(\gamma \lambda^ 2+\lambda +\gamma =0\) satisfies \(| \lambda^ k-1| > \epsilon k^{-\delta}\) for some \(\epsilon,\delta >0\) and all \(k\in {\mathbb{N}}\), then there exists a unique local analytic Riemann surface \(R\subset {\mathbb{C}}^ 2\) (a complex curve in \({\mathbb{C}}^ 2)\) with \(R\cap M=K=K^ 1\cup K^ 2\), \(K^ 1\) and \(K^ 2\) real curves going transversally through the origin. A generalization to higher dimensions is given for \(M\subset {\mathbb{C}}^ n\), \(\dim_{{\mathbb{R}}} M=n.\)
scientific article; zbMATH DE number 3981533