no. 6

Curvature restrictions for Levi-flat real hypersurfaces in complex projective planes
[Restrictions sur une courbure des hypersurfaces réelles Levi-plates dans plans projectifs complexes]
Annales de l'Institut Fourier, Tome 65 (2015) no. 6, pp. 2547-2569.

Nous étudions des restrictions sur une courbure des hypersurfaces réelles Levi-plates dans des plans projectifs complexes, dont l’existence est en question. Nous nous focalisons sur sa courbure de Ricci totalement réelle, c’est-â-dire la courbure de Ricci de l’hypersurface réelle dans la direction du champ de Reeb, et nous démontrons qu’elle ne peut pas être supérieure à -4 le long de l’hypersurface réelle Levi-plate. Nous nous appuyons sur un théorème de finitude pour l’espace des 2-formes holomorphes de carrés intégrables sur le complément de l’hypersurface réelle Levi-plate, où la courbure joue le rôle de la taille de l’holonomie infinitésimale de son feuilletage de Levi.

We study curvature restrictions of Levi-flat real hypersurfaces in complex projective planes, whose existence is in question. We focus on its totally real Ricci curvature, the Ricci curvature of the real hypersurface in the direction of the Reeb vector field, and show that it cannot be greater than -4 along a Levi-flat real hypersurface. We rely on a finiteness theorem for the space of square integrable holomorphic 2-forms on the complement of the Levi-flat real hypersurface, where the curvature plays the role of the size of the infinitesimal holonomy of its Levi foliation.

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DOI : https://doi.org/10.5802/aif.2995
Classification : 32V15,  32V40,  53B25,  53C12
Mots clés : hypersurface réelle Levi-plate, courbure de Ricci totalement réelle, formule d’adjonction, formule intégrale 
@article{AIF_2015__65_6_2547_0,
     author = {Adachi, Masanori and Brinkschulte, Judith},
     title = {Curvature restrictions for {Levi-flat} real hypersurfaces in complex projective planes},
     journal = {Annales de l'Institut Fourier},
     pages = {2547--2569},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {65},
     number = {6},
     year = {2015},
     doi = {10.5802/aif.2995},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2995/}
}
Adachi, Masanori; Brinkschulte, Judith. Curvature restrictions for Levi-flat real hypersurfaces in complex projective planes. Annales de l'Institut Fourier, Tome 65 (2015) no. 6, pp. 2547-2569. doi : 10.5802/aif.2995. https://aif.centre-mersenne.org/articles/10.5802/aif.2995/

[1] Adachi, M.; Brinkschulte, J. A global estimate for the Diederich–Fornaess index of weakly pseudoconvex domains (to appear in Nagoya Math. J.)

[2] Adachi, Masanori A local expression of the Diederich–Fornaess exponent and the exponent of conformal harmonic measures, Bull. Braz. Math. Soc. (N.S.), Tome 46 (2015) no. 1, pp. 65-79 | Article | MR 3324178

[3] Bejancu, Aurel; Deshmukh, Sharief Real hypersurfaces of CP n with non-negative Ricci curvature, Proc. Amer. Math. Soc., Tome 124 (1996) no. 1, pp. 269-274 | MR 1277096 | Zbl 0866.53041

[4] Camacho, C.; Lins Neto, A.; Sad, P. Minimal sets of foliations on complex projective spaces, Inst. Hautes Études Sci. Publ. Math., Tome 68 (1988), pp. 187-203 | Numdam | MR 1001454 | Zbl 0682.57012

[5] Cao, Jianguo; Shaw, Mei-Chi A new proof of the Takeuchi theorem, Lecture notes of Seminario Interdisciplinare di Matematica. Vol. IV, S.I.M. Dep. Mat. Univ. Basilicata, Potenza, 2005, pp. 65-72 | MR 2222537 | Zbl 1108.32013

[6] Cerveau, Dominique Minimaux des feuilletages algébriques de CP(n), Ann. Inst. Fourier (Grenoble), Tome 43 (1993) no. 5, pp. 1535-1543 | Numdam | MR 1275208 | Zbl 0803.32018

[7] Demailly, Jean-Pierre Estimations L 2 pour l’opérateur ∂ ¯ d’un fibré vectoriel holomorphe semi-positif au-dessus d’une variété kählérienne complète, Ann. Sci. École Norm. Sup. (4), Tome 15 (1982) no. 3, pp. 457-511 | Numdam | MR 690650 | Zbl 0507.32021

[8] Deroin, Bertrand Hypersurfaces Levi-plates immergées dans les surfaces complexes de courbure positive, Ann. Sci. École Norm. Sup. (4), Tome 38 (2005) no. 1, pp. 57-75 | Numdam | MR 2136481 | Zbl 1070.37031

[9] Fu, Siqi; Shaw, Mei-Chi The Diederich–Fornæss exponent and non-existence of Stein domains with Levi-flat boundaries (J. Geom. Anal., published online on 25 November 2014.)

[10] Goldberg, Samuel I.; Kobayashi, Shoshichi Holomorphic bisectional curvature, J. Differential Geom., Tome 1 (1967), pp. 225-233 | MR 227901 | Zbl 0169.53202

[11] Greene, R. E.; Wu, H. On Kähler manifolds of positive bisectional curvature and a theorem of Hartogs, Abh. Math. Sem. Univ. Hamburg, Tome 47 (1978), pp. 171-185 | MR 499318 | Zbl 0431.32017

[12] Griffiths, Phillip A. The extension problem in complex analysis. II. Embeddings with positive normal bundle, Amer. J. Math., Tome 88 (1966), pp. 366-446 | MR 206980 | Zbl 0147.07502

[13] Henkin, Gennadi M.; Iordan, Andrei Regularity of ∂ ¯ on pseudoconcave compacts and applications, Asian J. Math., Tome 4 (2000) no. 4, pp. 855-883 | MR 1870663 | Zbl 0998.32021

[14] Hörmander, Lars L 2 estimates and existence theorems for the ∂ ¯ operator, Acta Math., Tome 113 (1965), pp. 89-152 | MR 179443 | Zbl 0158.11002

[15] Iordan, Andrei; Matthey, Fanny Régularité de l’opérateur ∂ ¯ et théorème de Siu sur la non-existence d’hypersurfaces Levi-plates dans l’espace projectif complexe ℂℙ n , n≥3, C. R. Math. Acad. Sci. Paris, Tome 346 (2008) no. 7-8, pp. 395-400 | MR 2417557 | Zbl 1138.32021

[16] Lins Neto, Alcides A note on projective Levi flats and minimal sets of algebraic foliations, Ann. Inst. Fourier (Grenoble), Tome 49 (1999) no. 4, pp. 1369-1385 | Numdam | MR 1703092 | Zbl 0963.32022

[17] Matsumoto, Kazuko Levi form of logarithmic distance to complex submanifolds and its application to developability, Complex analysis in several variables—Memorial Conference of Kiyoshi Oka’s Centennial Birthday (Adv. Stud. Pure Math.) Tome 42, Math. Soc. Japan, Tokyo, 2004, pp. 203-207 | MR 2087052 | Zbl 1080.32035

[18] Ohsawa, Takeo Kählerity and pseudoconvexity (Abstracts for Complex Geometry 2010 (Mabuchi 60), Osaka, 2010, available at http://www.math.sci.osaka-u.ac.jp/~mabuchi/files/Ohsawa.pdf (in Japanese))

[19] Ohsawa, Takeo; Sibony, Nessim Bounded p.s.h. functions and pseudoconvexity in Kähler manifold, Nagoya Math. J., Tome 149 (1998), pp. 1-8 | MR 1619572 | Zbl 0911.32027

[20] Siu, Yum-Tong Nonexistence of smooth Levi-flat hypersurfaces in complex projective spaces of dimension ≥3, Ann. of Math. (2), Tome 151 (2000) no. 3, pp. 1217-1243 | MR 1779568 | Zbl 0980.53065

[21] Straube, Emil J. Lectures on the L 2 -Sobolev theory of the ∂ ¯-Neumann problem, ESI Lectures in Mathematics and Physics, European Mathematical Society (EMS), Zürich, 2010, viii+206 pages | MR 2603659 | Zbl 1247.32003

[22] Takeuchi, Akira Domaines pseudoconvexes infinis et la métrique riemannienne dans un espace projectif, J. Math. Soc. Japan, Tome 16 (1964), pp. 159-181 | MR 173789 | Zbl 0141.08804