On a global estimate of the Diederich--Fornaess index of Levi-flat real hypersurfaces

TL;DR

This paper reviews recent advances in understanding the Diederich--Fornaess index for complex domains, especially those with Levi-flat boundaries, highlighting a key relationship between holonomy and normal bundle curvature.

Contribution

It provides a comprehensive review of recent progress and establishes a new relation between holonomy norm and normal bundle curvature for Levi-flat hypersurfaces.

Findings

Holonomy norm exceeds normal bundle curvature for Levi-flat hypersurfaces

Review of recent progress in Diederich--Fornaess index studies

Emphasis on Levi-flat boundary cases

Abstract

In this expository paper, we review a recent progress of the study of the Diederich--Fornaess index of complex domains with emphasis on the case of domains with Levi-flat boundary. It is exhibited that for any compact Levi-flat real hypersurface, the norm of its infinitesimal holonomy must exceed the curvature of its normal bundle at a point.