The Diederich-Forn{\ae}ss exponent and non-existence of Stein domains with Levi-flat boundaries

TL;DR

This paper investigates the Diederich-Forn{ {a}}ss exponent's role in the non-existence of Stein domains with Levi-flat boundaries, establishing a link between the exponent's value and boundary Levi-form rank in complex manifolds.

Contribution

It establishes a new relation between the Diederich-Forn{ {a}}ss exponent and the Levi-form rank at boundary points, providing conditions for the non-existence of certain Stein domains.

Findings

If the Diederich-Forn{ {a}}}ss exponent exceeds k/n, the boundary has a point with Levi-form rank at least k.

The paper proves that Stein domains with high Diederich-Forn{ {a}}ss exponents cannot have Levi-flat boundaries.

It links geometric boundary properties to the analytic Diederich-Forn{ {a}}ss exponent in complex manifolds.

Abstract

We study the Diederich-Forn{\ae}ss exponent and relate it to non-existence of Stein domains with Levi-flat boundaries in complex manifolds. In particular, we prove that if the Diederich-Forn{\ae}ss exponent of a smooth bounded Stein domain in an $n$-dimensional complex manifold is $>k/n$, then it has a boundary point at which the Levi-form has rank $\ge k$.