Convex domains with locally Levi-flat boundaries

TL;DR

The paper proves that certain convex domains with smooth Levi-flat boundaries in complex space are locally equivalent to a product of a planar domain and complex Euclidean space, highlighting limitations when the Levi form's rank varies.

Contribution

It establishes a local equivalence for convex domains with smooth Levi-flat boundaries and clarifies the boundary conditions where this equivalence holds or fails.

Findings

Convex domains with smooth Levi-flat boundaries are locally equivalent to a product space.

This equivalence does not hold when the Levi form's rank is only locally constant for some k<n-1.

The result clarifies the geometric structure of such domains in complex analysis.

Abstract

It is shown that a domain in $\C^n$ which is locally convex and has $\mathcal C^1$-smooth Levi-flat boundary is locally linearly equivalent to a Cartesian product of a planar domain and $\C^{n-1}.$ This result does not extend to the case where the Levi form has locally constant rank for some $k<n-1$.