Flattening of CR singular points and analyticity of local hull of holomorphy

TL;DR

This paper investigates conditions under which real analytic submanifolds in complex space can be locally flattened and extended holomorphically, providing new results for points with elliptic directions and formal flattening under certain assumptions.

Contribution

It offers an affirmative answer to the holomorphic flattening problem at complex tangent points with elliptic directions and establishes a formal flattening theorem assuming a non-parabolic direction.

Findings

Confirmed flattening at points with elliptic directions.

Established a formal flattening theorem under non-parabolic assumptions.

Connected flattening to the local hull of holomorphy of the submanifold.

Abstract

A primary goal in this paper is to study the question that asks when a real analytic submanifold $M$ in ${\mathbb{C}}^{n+1}$ bounds a real analytic (up to $M$) Levi-flat hypersurface $\hat{M}$ near $p\in M$ such that $\hat{M}$ is foliated by a family of complex hypersurfaces moving along the normal direction of $M$ at $p$, and gives the invariant local hull of holomorphy of $M$ near $p$. This question is equivalent to the holomorphic flattening problem for $M$ near $p$. We will give an affirmative answer to above question when $p$ is a real complex tangent point with at least one elliptic direction (positively curved direction). We also obtain a formal flattening theorem under the assumption of one non-parabolic direction.