The core of the Levi distribution

Gian Maria Dall’Ara; Samuele Mongodi

doi:10.5802/jep.239

The core of the Levi distribution
[Le cœur de la distribution de Levi]

Gian Maria Dall’Ara ¹ ; Samuele Mongodi ²

¹ Istituto Nazionale di Alta Matematica “F. Severi”, Research Unit Scuola Normale Superiore Piazza dei Cavalieri, 7, 56126 Pisa, Italy
² Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano-Bicocca Via Roberto Cozzi, 5, 20125 Milano, Italy

Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 1047-1095.

Résumé
Abstract

Nous introduisons un nouvel invariant géométrique des variétés CR de type hypersurface, que nous appelons le « cœur de Levi » de la variété. Lorsque la variété est le bord d’un domaine pseudoconvexe lisse et borné, nous montrons comment le cœur de Levi est lié à deux autres invariants globaux importants en plusieurs variables complexes : l’indice de Diederich-Fornæss et la classe de D’Angelo (à savoir l’ensemble des formes de D’Angelo du bord). Nous montrons également que le cœur de Levi est trivial lorsque le domaine est de type fini au sens de D’Angelo, ou que l’ensemble des points faiblement pseudoconvexes est contenu dans une sous-variété totalement réelle, alors qu’il n’est pas trivial si le bord contient un ensemble de maxima locaux. Comme corollaires à la théorie développée ici, nous prouvons que pour tout domaine pseudoconvexe borné et lisse avec un cœur de Levi trivial, l’indice de Diederich-Fornæss est égal à $1$ et le problème de $\bar{\partial}$ -Neumann est exactement régulier (via un résultat de Kohn et sa généralisation par Harrington). Notre travail s’appuie sur des résultats récents de Liu et Adachi-Yum et les étend.

We introduce a new geometrical invariant of CR manifolds of hypersurface type, which we call 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–Fornæss 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–Fornæss index is $1$ and the $\bar{\partial}$ -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.

Reçu le : 2022-03-17
Accepté le : 2023-04-20
Publié le : 2023-05-25

DOI : 10.5802/jep.239

Classification : 32T20, 32T27, 32W05, 32V15
Keywords: Weakly pseudoconvex domains, Diederich-Fornæss index, CR manifolds, geometric invariants of pseudoconvex domains, Levi form
Mot clés : Domaines faiblement pseudoconvexes, indice de Diederich-Fornæss, variétés CR, invariants géométriques de domaines pseudoconvexes, forme de Levi

Affiliations des auteurs :

Gian Maria Dall’Ara ¹ ; Samuele Mongodi ²

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {The core of the {Levi} distribution},
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Gian Maria Dall’Ara; Samuele Mongodi. The core of the Levi distribution. Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 1047-1095. doi : 10.5802/jep.239. https://jep.centre-mersenne.org/articles/10.5802/jep.239/

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