The core of the Levi distribution

Dall’Ara, Gian Maria; Mongodi, Samuele

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

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 $\overline{\partial }$-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.

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 $\overline{\partial }$-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.

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

[AB15] Adachi, Masanori; Brinkschulte, Judith A global estimate for the Diederich-Fornæss index of weakly pseudoconvex domains, Nagoya Math. J., Volume 220 (2015), pp. 67-80 | Zbl | DOI

[AY21] Adachi, Masanori; Yum, Jihun Diederich-Fornæss and Steinness indices for abstract CR manifolds, J. Geom. Anal., Volume 31 (2021) no. 8, pp. 8385-8413 | DOI | Zbl

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

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

[BDD07] Barletta, Elisabetta; Dragomir, Sorin; Duggal, Krishan L. Foliations in Cauchy-Riemann geometry, Math. Surveys and Monographs, 140, American Mathematical Society, Providence, RI, 2007 | DOI

[BM22] Bianchi, Fabrizio; Mongodi, Samuele On minimal kernels and Levi currents on weakly complete complex manifolds, Proc. Amer. Math. Soc., Volume 150 (2022) no. 9, pp. 3927-3939 | MR | Zbl | DOI

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

[BS93] Boas, Harold P.; Straube, Emil J. de Rham cohomology of manifolds containing the points of infinite type, and Sobolev estimates for the $\overline{\partial }$-Neumann problem, J. Geom. Anal., Volume 3 (1993) no. 3, pp. 225-235 | Zbl | MR | DOI

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

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

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

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

[CLN85] Camacho, César; Lins Neto, Alcides Geometric theory of foliations, Birkhäuser Boston, Inc., Boston, MA, 1985 | DOI

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

[DF77] Diederich, Klas; Fornæss, John Erik Pseudoconvex domains: bounded strictly plurisubharmonic exhaustion functions, Invent. Math., Volume 39 (1977) no. 2, pp. 129-141 | Zbl | MR | DOI

[DF78] Diederich, Klas; Fornæss, John Erik Pseudoconvex domains with real-analytic boundary, Ann. of Math. (2), Volume 107 (1978) no. 2, pp. 371-384 | MR | Zbl | DOI

[DT06] Dragomir, Sorin; Tomassini, Giuseppe Differential geometry and analysis on CR manifolds, Progress in Math., 246, Birkhäuser Boston, Inc., Boston, MA, 2006

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

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

[FLT08] Forstnerič, Franc; Laurent-Thiébaut, Christine Stein compacts in Levi-flat hypersurfaces, Trans. Amer. Math. Soc., Volume 360 (2008) no. 01, pp. 307-330 | MR | Zbl | DOI

[For81] Forster, Otto Lectures on Riemann Surfaces, Springer, New York, 1981 | DOI

[FS16] Fu, Siqi; Shaw, Mei-Chi The Diederich-Fornæss exponent and non-existence of Stein domains with Levi-flat boundaries, J. Geom. Anal., Volume 26 (2016) no. 1, pp. 220-230 | Zbl | DOI

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

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

[KLP18] Krantz, Steven George; Liu, Bingyuan; Peloso, Marco Maria Geometric analysis on the Diederich-Fornæss index, J. Korean Math. Soc., Volume 55 (2018) no. 4, pp. 897-921 | DOI | Zbl

[KN65] Kohn, Joseph J.; Nirenberg, Louis Non-coercive boundary value problems, Comm. Pure Appl. Math., Volume 18 (1965), pp. 443-492 | DOI | Zbl | MR

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

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

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

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

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

[LM87] Libermann, Paulette; Marle, Charles-Michel Symplectic geometry and analytical mechanics, Math. and its Applications, 35, D. Reidel Publishing Co., Dordrecht, 1987 | DOI

[LS22] Liu, Bingyuan; Straube, Emil J. Diederich-Fornæss index and global regularity in the $\overline{\partial }$-Neumann problem: domains with comparable Levi eigenvalues, 2022 | arXiv

[MT15] Mongodi, Samuele; Tomassini, Giuseppe Transversally pseudoconvex semiholomorphic foliations, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Mem. (9) Mat. Appl., Volume 26 (2015) no. 1, pp. 23-36 | DOI | MR | Zbl

[MT16] Mongodi, Samuele; Tomassini, Giuseppe 1-complete semiholomorphic foliations, Trans. Amer. Math. Soc., Volume 368 (2016) no. 9, pp. 6271-6292 | DOI | MR | Zbl

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

[OS98] Ohsawa, Takeo; Sibony, Nessim Bounded p.s.h. functions and pseudoconvexity in Kähler manifold, Nagoya Math. J., Volume 149 (1998), pp. 1-8 | DOI | Zbl

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

[Sib17] Sibony, Nessim Levi problem in complex manifolds, Math. Ann., Volume 371 (2017) no. 3-4, pp. 1047-1067 | DOI | Zbl | MR

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

[SS02] Straube, Emil J.; Sucheston, Marcel K. Plurisubharmonic defining functions, good vector fields, and exactness of a certain one form, Monatsh. Math., Volume 136 (2002) no. 3, pp. 249-258 | DOI | MR | Zbl

[SS03] Straube, Emil J.; Sucheston, Marcel K. Levi foliations in pseudoconvex boundaries and vector fields that commute approximately with $\overline{\partial }$, Trans. Amer. Math. Soc., Volume 355 (2003) no. 1, pp. 143-154 | DOI | Zbl | MR

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

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

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

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

[Yum21] Yum, Jihun CR-invariance of the Steinness index, Math. Z., Volume 297 (2021) no. 3-4, pp. 1043-1056 | Zbl | MR