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

TL;DR

The paper analyzes local polynomial convexity for real surfaces in $\mathbb{C}^2$ with an isolated CR singularity of cubic order, introducing the one-parameter family $M_t$ and its normal form. By pulling back surfaces via a unique proper holomorphic map to unions of three totally-real planes, it reduces questions of convexity to Weinstock's normal form and applies Kallin's lemma, Maslov-type index, and Wiegerinck-type criteria to obtain sharp thresholds: $M_t$ is not locally polynomially convex for $t<1$, its local hull contains a ball for $t<\sqrt{3}/2$, and it is locally polynomially convex for $t\ge\sqrt{3/2}$; for $\sqrt{3}/2\le t<1$ the hull contains a 1-parameter family of analytic discs through the origin. Removing higher-order terms raises the convexity threshold to $t\ge\dfrac{\sqrt{15-\sqrt{33}}}{2\sqrt{2}}$, and the authors also derive new results on the local polynomial convexity of unions of three totally-real planes and extend some conclusions to unions of three totally-real surfaces. The work advances a Bishop-type dichotomy in the nonreal lowest-order context and provides tools for understanding hulls and analytic discs near higher-order CR singularities.

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 $M_t\cap \overline{B(0;δ)}$ contains a one parameter family of analytic discs passing through the origin for every $δ>0$. 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.

Theorems & Definitions (36)

• Definition 1.3
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9
• Theorem 1.10
• Theorem 1.11
• Remark 1.12
• Lemma 2.1
• proof