The core of the Levi distribution

TL;DR

This paper introduces the Levi core, a new geometric invariant of CR manifolds, and explores its relation to key invariants in complex analysis, revealing conditions for triviality and nontriviality with implications for the Diederich--Forn{ }e}ss index and regularity of the $ar{d}$-Neumann problem.

Contribution

It defines the Levi core invariant for CR manifolds and establishes its connections to the Diederich--Forn{ }e}ss index and boundary regularity, expanding understanding of geometric properties in complex analysis.

Findings

Levi core is trivial for finite-type domains.

Levi core is nontrivial if boundary contains a local maximum set.

Domains with trivial Levi core have Diederich--Forn{ }e}ss index equal to one.

Abstract

We introduce a new geometrical invariant of CR manifolds of hypersurface type, which we dub the "Levi core" of the manifold. When the manifold is the boundary of a smooth bounded pseudoconvex domain, we show how the Levi core is related to two other important global invariants in several complex variables: the Diederich--Forn{\ae}ss index and the D'Angelo class (namely the set of D'Angelo forms of the boundary). We also show that the Levi core is trivial whenever the domain is of finite-type in the sense of D'Angelo, or the set of weakly pseudoconvex points is contained in a totally real submanifold, while it is nontrivial if the boundary contains a local maximum set. As corollaries to the theory developed here, we prove that for any smooth bounded pseudoconvex domain with trivial Levi core the Diederich--Forn{\ae}ss index is one and the $\overline{\partial}$-Neumann problem is exactly regular (via a result of Kohn and its generalization by Harrington). Our work builds on and expands recent results of Liu and Adachi--Yum.