A survey on Levi-flats and related topics

Some information for a mini lecture course that was held at Universität zu Köln in February 2023.

This mini lecture course was finished. Thank you for your interest and asking many questions. Any further questions and/or comments are very welcomed. Please feel free to write an email to me!

Abstract

Levi-flats are smooth manifolds foliated by complex manifolds. They naturally arise in some problems on several complex variables and theory of foliations. We try to give a survey on their origin, basic examples, fundamental results (in global and non-singular setting) and open problems.

Schedule

1. (07.02. um 14.00 Uhr) Interpolation theorem in one variable, Hartogs’ phenomenon, notion of pseudoconvexity, Oka’s lemma and the solution of classical Levi problem by $L^2$ method (Part 1).
2. (08.02. um 14.00 Uhr) The counterexample of Grauert (Part 1), definition of foliation and Levi-flat CR manifold and statements of Inaba’s theorem and Ohsawa-Sibony embedding (Part 2). Proof of Inaba’s theorem was omitted.
3. (09.02. um 10.00 Uhr) Proof of Ohsawa-Sibony embedding (Part 2).
4. (14.02. um 14.00 Uhr) Suspension, examples by Diederich-Fornaess and Diederich-Ohsawa, idea of the proof for Diederich-Ohsawa theorem, specific disc bundle without non-constant bounded holomorphic functions (Part 3).
5. (15.02. um 14.00 Uhr) Remarks for the specific disc bundle (Part 3). The Diederich-Fornaess index. Upper bound of the Diederich-Fornaess index for domains with Levi-flat boundary. Construction of ddbar-closed current for the specific disc bundle (Part B). Part A was omitted.

Plan

1. Levi-flats as counterexamples in the Levi problem

We first recall the notion of pseudoconvexity and quickly review the affirmative solution to the classical Levi problem using $L^2$ method. Then we exhibit some examples of Levi-flats as counterexamples to the Levi problem in complex manifolds.

2. Function theory on Levi-flats

We discuss compact Levi-flats as abstract CR manifolds and study function theory on them. For transversely smooth CR functions/sections, the situation is similar to that on compact complex manifolds. We sketch proofs for Inaba’s theorem and Ohsawa-Sibony’s embedding theorem.

3. Disc bundles

Given $PSL(2,\mathbb{R})$-representations of the fundamental group of a complex manifold, we obtain examples of Levi-flats via suspension construction. By these examples we observe how ergodic property of the representation affects pseudoconvexity of the domain bounded by Levi-flats and function theory on the Levi-flat CR manifolds.

If we have more time, I can continue with additional lectures:

A. (Non-)existence problem

Examples of Levi-flats with 1-convex complements are very rare. For instance, it is conjectured that there is no smoothly bounded domain with Levi-flat boundary in the complex projective plane. We review known results around this problem.

B. Diederich-Fornaess index

The Diederich-Fornaess index is a way to measure the strength of hyperconvexity for weakly pseudoconvex domains. Although this is a classical topic since late 1970s, its geometric or quantitative study has attracted attention since early 2010s. We review recent development around this topic.