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

TL;DR

This paper constructs regular Stein neighborhoods around unions of two maximally totally real subspaces in complex Euclidean space, under smallness conditions on a defining matrix, with applications to immersions of real manifolds.

Contribution

It provides a new method to construct Stein neighborhoods of unions of totally real subspaces with small defining matrices, extending to applications in real manifold immersions.

Findings

Constructed strongly pseudoconvex neighborhoods around unions of totally real subspaces.

Proved deformation retraction of neighborhoods onto the union of subspaces.

Applied results to totally real immersions with finitely many double points.

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.