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

TL;DR

This paper classifies certain complex surfaces with isolated singularities in ^2, analyzing their polynomial convexity properties and revealing conditions under which their local hulls contain analytic discs.

Contribution

It introduces a parametric family of surfaces with isolated CR singularities and determines their local polynomial convexity and hull properties, including new results on unions of totally-real planes.

Findings

Surfaces are reducible to a one-parameter family under certain conditions.

Non-polynomial convexity occurs for t<1, with hulls containing balls for t<rac{rac{3}{2}}.

For rac{rac{3}{2}}{rac{2}{2}}, the surfaces are locally polynomially convex.

Abstract

Under certain geometric condition, the surfaces in $\mathbb{C}^2$ with isolated CR singularity at the origin and with cubic lowest degree homogeneous term in its graph near the origin, can be reduced, up to biholomorphism of $\mathbb{C}^2$, to a one parameter family of the form \[ M_t:=\left\{(z,w)\in\mathbb{C}^2: w=z^2\overline{z}+tz\overline{z}^2+\dfrac{t^2}{3} \overline{z}^3+o(|z|^3)\right\},\;\; t\in (0,\infty) \] near the origin. We prove that $M_t$ is not locally polynomially convex if $t<1$. The local hull contains a ball centred at the origin if $t<\sqrt{3}/2$. We also prove that $M_t$ is locally polynomially convex for $t\geq\sqrt{\dfrac{3}{2}}$. We show that, for $\sqrt{3}/2\leq t<1$, the polynomial hull of $M_t\cap \overline{B(0;\delta)}$ contains a one parameter family of analytic discs passing through the origin for every $\delta>0$. We also prove that, if we remove the higher order terms from the graphing function of $M_t$, it is locally polynomially convex for $t\geq\dfrac{\sqrt{15-\sqrt{33}}}{2\sqrt{2}}$. Some new results about the local polynomial convexity of the union of three totally-real planes are also reported.