Certain real surfaces in ${\mathbb{C}}^2$ with degenerated CR singularities

Abstract

In this paper, we study the local polynomial convexity of certain smooth real surfaces in \(\mathbb{C}^2\) with isolated CR singularity at the origin with higher-order of degeneracy. Under the assumption that the surface can be pulled back to a union of finitely many pairwise transverse totally real surfaces by a proper holomorphic map from ${\mathbb{C}}^2$ to ${\mathbb{C}}^2$, we obtain a normal form for such surfaces near the origin as $\{(z,w)\in {\mathbb{C}}^2:w={\overline{z}}^k+o(|z{|}^k)\}$ or $M_t:=\left\{(z,w)\in {\mathbb{C}}^2:w=(z+t\overline{z}{)}^k+o(|z{|}^k)\right\}$, for some \(t>0\), where the parameter $t$ is a local biholomorphic invariant. We focus on the surfaces with order of degeneracy $k\ge 3$. We prove that $M_t$ is locally polynomially convex at the origin if $t>cosec\left(\frac{\ddot{I}}{€}k\right)$. On the other hand, for $0<t<\frac{1}{k-2}$, we will also show that $M_t$ fails to be locally polynomially convex at the origin; and furthermore, a $(2k-3)$-parameter family of analytic discs attached to $M_t$ for $0<t<\min \left\{\sin \left(\frac{\ddot{I}}{€}k\right),\frac{1}{k-2}\right\}$.

22 pages, comments are welcome