Flattening a non-degenerate CR singular point of real codimension two

TL;DR

This paper advances the understanding of CR singular points of real codimension two by establishing a general flattening theorem, solving the local complex Plateau problem, and proving analyticity of the local hull of holomorphy.

Contribution

It introduces a comprehensive flattening theorem for non-degenerate CR singular points, extending previous geometric and formal approaches.

Findings

Provides a general flattening theorem for CR singular points.

Solves the local complex Plateau problem in this context.

Establishes the analyticity of the local hull of holomorphy near CR singular points.

Abstract

This paper continues the previous studies in two papers of Huang-Yin [HY3-4] on the flattening problem of a CR singular point of real codimension two sitting in a submanifold in ${\mathbb C}^{n+1}$ with $n+1\ge 3$, whose CR points are non-minimal. Partially based on the geometric approach initiated in [HY3] and a formal theory approach used in [HY4], we are able to provide a very general flattening theorem for a non-degenerate CR singular point. As an application, we provide a solution to the local complex Plateau problem and obtain the analyticity of the local hull of holomorphy near a real analytic definite CR singular point in a general setting.