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

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

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{JEP_2023__10__1047_0,
     author = {Gian Maria Dall{\textquoteright}Ara and Samuele Mongodi},
     title = {The core of the {Levi} distribution},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {1047--1095},
     publisher = {\'Ecole polytechnique},
     volume = {10},
     year = {2023},
     doi = {10.5802/jep.239},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.239/}
}

TY  - JOUR
AU  - Gian Maria Dall’Ara
AU  - Samuele Mongodi
TI  - The core of the Levi distribution
JO  - Journal de l’École polytechnique — Mathématiques
PY  - 2023
SP  - 1047
EP  - 1095
VL  - 10
PB  - École polytechnique
UR  - https://jep.centre-mersenne.org/articles/10.5802/jep.239/
DO  - 10.5802/jep.239
LA  - en
ID  - JEP_2023__10__1047_0
ER  -

%0 Journal Article
%A Gian Maria Dall’Ara
%A Samuele Mongodi
%T The core of the Levi distribution
%J Journal de l’École polytechnique — Mathématiques
%D 2023
%P 1047-1095
%V 10
%I École polytechnique
%U https://jep.centre-mersenne.org/articles/10.5802/jep.239/
%R 10.5802/jep.239
%G en
%F JEP_2023__10__1047_0

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/

Bibliographie
Cité par

[AB15] M. Adachi & J. Brinkschulte - “A global estimate for the Diederich-Fornæss index of weakly pseudoconvex domains”, Nagoya Math. J. 220 (2015), p. 67-80 | DOI | Zbl

[AY21] M. Adachi & J. Yum - “Diederich-Fornæss and Steinness indices for abstract CR manifolds”, J. Geom. Anal. 31 (2021) no. 8, p. 8385-8413 | DOI | Zbl

[Bar84] D. E. Barrett - “Irregularity of the Bergman projection on a smooth bounded domain in $C^{2}$ ”, Ann. of Math. (2) 119 (1984) no. 2, p. 431-436 | DOI | MR | Zbl

[Bar98] D. E. Barrett - “The Bergman projection on sectorial domains”, in Operator theory for complex and hypercomplex analysis (Mexico City, 1994), Contemp. Math., vol. 212, American Mathematical Society, Providence, RI, 1998, p. 1-24 | DOI | MR | Zbl

[BDD07] E. Barletta, S. Dragomir & K. L. Duggal - Foliations in Cauchy-Riemann geometry, Math. Surveys and Monographs, vol. 140, American Mathematical Society, Providence, RI, 2007 | DOI

[BM22] F. Bianchi & S. Mongodi - “On minimal kernels and Levi currents on weakly complete complex manifolds”, Proc. Amer. Math. Soc. 150 (2022) no. 9, p. 3927-3939 | DOI | MR | Zbl

[Bog91] A. Boggess - CR manifolds and the tangential Cauchy-Riemann complex, Studies in Advanced Math., CRC Press, Boca Raton, FL, 1991 | MR

[BS93] H. P. Boas & E. J. Straube - “de Rham cohomology of manifolds containing the points of infinite type, and Sobolev estimates for the $\bar{\partial}$ -Neumann problem”, J. Geom. Anal. 3 (1993) no. 3, p. 225-235 | DOI | MR | Zbl

[Cat84a] D. W. Catlin - “Boundary invariants of pseudoconvex domains”, Ann. of Math. (2) 120 (1984) no. 3, p. 529-586 | DOI | MR | Zbl

[Cat84b] D. W. Catlin - “Global regularity of the $\bar{\partial}$ -Neumann problem”, in Complex analysis of several variables (Madison, Wis., 1982), Proc. Sympos. Pure Math., vol. 41, American Mathematical Society, Providence, RI, 1984, p. 39-49 | DOI | MR | Zbl

[Chr96] M. Christ - “Global $C^{\infty}$ irregularity of the $\bar{\partial}$ -Neumann problem for worm domains”, J. Amer. Math. Soc. 9 (1996) no. 4, p. 1171-1185 | DOI | MR | Zbl

[Cie97] K. Ciesielski - Set theory for the working mathematician, London Math. Society Student Texts, vol. 39, Cambridge University Press, Cambridge, 1997 | DOI

[CLN85] C. Camacho & A. Lins Neto - Geometric theory of foliations, Birkhäuser Boston, Inc., Boston, MA, 1985 | DOI

[Cor96] P. D. Cordaro - “Global hypoellipticity for on certain compact three dimensional CR manifolds”, Resenhas 2 (1996) no. 4, p. 353-361 | MR | Zbl

[DF77] K. Diederich & J. E. Fornæss - “Pseudoconvex domains: bounded strictly plurisubharmonic exhaustion functions”, Invent. Math. 39 (1977) no. 2, p. 129-141 | DOI | MR | Zbl

[DF78] K. Diederich & J. E. Fornæss - “Pseudoconvex domains with real-analytic boundary”, Ann. of Math. (2) 107 (1978) no. 2, p. 371-384 | DOI | MR | Zbl

[DT06] S. Dragomir & G. Tomassini - Differential geometry and analysis on CR manifolds, Progress in Math., vol. 246, Birkhäuser Boston, Inc., Boston, MA, 2006

[D’A79] J. P. D’Angelo - “Finite type conditions for real hypersurfaces”, J. Differential Geom. 14 (1979) no. 1, p. 59-66 | MR | Zbl

[D’A87] J. P. D’Angelo - “Iterated commutators and derivatives of the Levi form”, in Complex analysis (University Park, Pa., 1986), Lect. Notes in Math., vol. 1268, Springer, Berlin, 1987, p. 103-110 | DOI | MR | Zbl

[FLT08] F. Forstnerič & C. Laurent-Thiébaut - “Stein compacts in Levi-flat hypersurfaces”, Trans. Amer. Math. Soc. 360 (2008) no. 01, p. 307-330 | DOI | MR | Zbl

[For81] O. Forster - Lectures on Riemann Surfaces, Springer, New York, 1981 | DOI

[FS16] S. Fu & M.-C. Shaw - “The Diederich-Fornæss exponent and non-existence of Stein domains with Levi-flat boundaries”, J. Geom. Anal. 26 (2016) no. 1, p. 220-230 | DOI | Zbl

[Har11] P. S. Harrington - “Global regularity for the $\bar{\partial}$ -Neumann operator and bounded plurisubharmonic exhaustion functions”, Adv. Math. 228 (2011) no. 4, p. 2522-2551 | DOI | MR | Zbl

[HS93] R. Hurri-Syrjänen - “The John-Nirenberg inequality and a Sobolev inequality in general domains”, J. Math. Anal. Appl. 175 (1993) no. 2, p. 579-587 | DOI | MR | Zbl

[KLP18] S. G. Krantz, B. Liu & M. M. Peloso - “Geometric analysis on the Diederich-Fornæss index”, J. Korean Math. Soc. 55 (2018) no. 4, p. 897-921 | DOI | Zbl

[KN65] J. J. Kohn & L. Nirenberg - “Non-coercive boundary value problems”, Comm. Pure Appl. Math. 18 (1965), p. 443-492 | DOI | MR | Zbl

[Koh79] J. J. Kohn - “Subellipticity of the $\bar{\partial}$ -Neumann problem on pseudo-convex domains: sufficient conditions”, Acta Math. 142 (1979) no. 1-2, p. 79-122 | DOI | MR | Zbl

[Koh99] J. J. Kohn - “Quantitative estimates for global regularity”, in Analysis and geometry in several complex variables (Katata, 1997), Trends Math., Birkhäuser Boston, Boston, MA, 1999, p. 97-128 | DOI | Zbl

[Lav18] S. Lavau - “A short guide through integration theorems of generalized distributions”, Differential Geom. Appl. 61 (2018), p. 42-58 | DOI | MR | Zbl

[Liu19a] B. Liu - “The Diederich-Fornæss index I: for domains of non-trivial index”, Adv. Math. 353 (2019), p. 776-801 | DOI | MR | Zbl

[Liu19b] B. Liu - “The Diederich-Fornæss index II: For domains of trivial index”, Adv. Math. 344 (2019), p. 289-310 | DOI | MR | Zbl

[LM87] P. Libermann & C.-M. Marle - Symplectic geometry and analytical mechanics, Math. and its Applications, vol. 35, D. Reidel Publishing Co., Dordrecht, 1987 | DOI

[LS22] B. Liu & E. J. Straube - “Diederich-Fornæss index and global regularity in the $\bar{\partial}$ -Neumann problem: domains with comparable Levi eigenvalues”, 2022 | arXiv

[MT15] S. Mongodi & G. Tomassini - “Transversally pseudoconvex semiholomorphic foliations”, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Mem. (9) Mat. Appl. 26 (2015) no. 1, p. 23-36 | DOI | MR | Zbl

[MT16] S. Mongodi & G. Tomassini - “1-complete semiholomorphic foliations”, Trans. Amer. Math. Soc. 368 (2016) no. 9, p. 6271-6292 | DOI | MR | Zbl

[NGSdS03] J. A. Navarro González & J. B. Sancho de Salas - $C^{\infty}$ -differentiable spaces, Lect. Notes in Math., vol. 1824, Springer-Verlag, Berlin, 2003 | DOI

[OS98] T. Ohsawa & N. Sibony - “Bounded p.s.h. functions and pseudoconvexity in Kähler manifold”, Nagoya Math. J. 149 (1998), p. 1-8 | DOI | Zbl

[Shi64] K. Shiga - “Some aspects of real-analytic manifolds and differentiable manifolds”, J. Math. Soc. Japan 16 (1964), p. 128-142 | DOI | MR | Zbl

[Sib17] N. Sibony - “Levi problem in complex manifolds”, Math. Ann. 371 (2017) no. 3-4, p. 1047-1067 | DOI | MR | Zbl

[Slo86] Z. Slodkowski - “Local maximum property and $q$ -plurisubharmonic functions in uniform algebras”, J. Math. Anal. Appl. 115 (1986) no. 1, p. 105-130 | DOI | MR | Zbl

[SS02] E. J. Straube & M. K. Sucheston - “Plurisubharmonic defining functions, good vector fields, and exactness of a certain one form”, Monatsh. Math. 136 (2002) no. 3, p. 249-258 | DOI | MR | Zbl

[SS03] E. J. Straube & M. K. Sucheston - “Levi foliations in pseudoconvex boundaries and vector fields that commute approximately with $\bar{\partial}$ ”, Trans. Amer. Math. Soc. 355 (2003) no. 1, p. 143-154 | DOI | MR | Zbl

[Str08] E. J. Straube - “A sufficient condition for global regularity of the $\bar{\partial}$ -Neumann operator”, Adv. Math. 217 (2008) no. 3, p. 1072-1095 | DOI | MR | Zbl

[Str10] E. J. Straube - Lectures on the $ℒ^{2}$ -Sobolev theory of the $\bar{\partial}$ -Neumann problem, ESI Lectures in Mathematics and Physics, European Mathematical Society, Zürich, 2010 | DOI

[Tre22] J. N. Treuer - “Sufficient condition for compactness of the $\bar{\partial}$ -Neumann operator using the Levi core”, 2022 | arXiv

[Yum19] J. Yum - “On the Steinness index”, J. Geom. Anal. 29 (2019) no. 2, p. 1583-1607 | DOI | MR | Zbl

[Yum21] J. Yum - “CR-invariance of the Steinness index”, Math. Z. 297 (2021) no. 3-4, p. 1043-1056 | MR | Zbl

Cité par Sources :