On regular Stein neighborhoods of a union of two totally real planes in $\mathbb{C}^2$

TL;DR

This paper constructs regular Stein neighborhoods around the union of two totally real planes in ^2, using a local function  with Levi pseudoconvex sublevel sets, under small matrix perturbations.

Contribution

It provides a method to find regular Stein neighborhoods for unions of totally real planes with small perturbations in .

Findings

Constructed  with Levi pseudoconvex sublevel sets

Neighborhoods deformation retract to the union of planes

Applicable for small matrix entries in 

Abstract

In this paper we find regular Stein neighborhoods for a union of totally real planes $M=(A+iI)\mathbb{R}^2$ and $N=\mathbb{R}^2$ in $\mathbb{C}^2$ provided that the entries of a real $2 \times 2$ matrix $A$ are sufficiently small. A key step in our proof is a local construction of a suitable function $\rho$ near the origin. The sublevel sets of $\rho$ are strongly Levi pseudoconvex and admit strong deformation retraction to $M\cup N$.