A characterization of domains in $\C^n$ with locally Levi-flat boundaries

TL;DR

This paper characterizes domains in complex n-space with Levi-flat boundaries using boundary behavior of intrinsic metrics and kernels, providing insights into their geometric structure, especially for intermediate Levi form ranks.

Contribution

It introduces a new characterization of Levi-flat boundary domains via Kobayashi and Bergman metrics, extending understanding of their boundary geometry.

Findings

Domains with Levi-flat boundaries are characterized by metric boundary behavior.

Results include cases with intermediate Levi form ranks.

Provides criteria for Levi-flatness based on intrinsic metrics.

Abstract

A domain in $\C^n$ with Levi-flat boundary near a given point is characterized in terms of the boundary behavior of the Kobayashi or Bergman metrics, or of the Bergman kernel. Some results are given in the case of intermediate values of the rank of the Levi form.