On regular Stein neighborhoods of a union of two maximal totally real subspaces in $\mathbb{C}^n$

Abstract

We present a construction of regular Stein neighborhoods of a union of maximally totally real subspaces $M=(A+iI)\mathbb{R}^n$ and $N=\mathbb{R}^n$ in $\mathbb{C}^n$, provided that the entries of a real $n \times n$ matrix $A$ are sufficiently small. Our proof is based on a local construction of a suitable plurisubharmonic function $\rho$ near the origin, such that the sublevel sets of $\rho$ are strongly pseudoconvex and admit strong deformation retraction to $M\cup N$. We also give the application of this result to totally real immersions of real $n$-manifolds in $\mathbb{C}^n$ with only finitely many double points, and such that the union of the tangent spaces at each intersection in some local coordinates coincides with $M\cup N$, described above.