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.

## Teaching materials

- References in PDF
- Typed notes will be uploaded here…

## Schedule

- (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).
- (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.
- (09.02. um 10.00 Uhr) Proof of Ohsawa-Sibony embedding (Part 2).
- (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).
- (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.