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]
Adachi, Masanori1, 2 ; Brinkschulte, Judith3
1 Nagoya University, Graduate School of Mathematics, Nagoya 464-8602 (Japan)
2 Pohang University of Science and Technology, Center for Geometry and its Applications, Pohang 790-784 (Republic of Korea)
3 Universität Leipzig, Mathematisches Institut, PF 100920, D-04009 Leipzig (Germany)
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.

DOI : 10.5802/aif.2995
Classification : 32V15, 32V40, 53B25, 53C12
Keywords: Levi-flat real hypersurface, totally real Ricci curvature, adjunction formula, integral formula
Mot clés : hypersurface réelle Levi-plate, courbure de Ricci totalement réelle, formule d’adjonction, formule intégrale

Adachi, Masanori 1, 2 ; Brinkschulte, Judith 3

1 Nagoya University, Graduate School of Mathematics, Nagoya 464-8602 (Japan)
2 Pohang University of Science and Technology, Center for Geometry and its Applications, Pohang 790-784 (Republic of Korea)
3 Universität Leipzig, Mathematisches Institut, PF 100920, D-04009 Leipzig (Germany) 
@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/}
}
TY  - JOUR
AU  - Adachi, Masanori
AU  - Brinkschulte, Judith
TI  - Curvature restrictions for Levi-flat real hypersurfaces in complex projective planes
JO  - Annales de l'Institut Fourier
PY  - 2015
SP  - 2547
EP  - 2569
VL  - 65
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2995/
DO  - 10.5802/aif.2995
LA  - en
ID  - AIF_2015__65_6_2547_0
ER  - 
%0 Journal Article
%A Adachi, Masanori
%A Brinkschulte, Judith
%T Curvature restrictions for Levi-flat real hypersurfaces in complex projective planes
%J Annales de l'Institut Fourier
%D 2015
%P 2547-2569
%V 65
%N 6
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.2995/
%R 10.5802/aif.2995
%G en
%F AIF_2015__65_6_2547_0
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.), Volume 46 (2015) no. 1, pp. 65-79 | DOI | MR

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

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

[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 | Zbl

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

[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), Volume 15 (1982) no. 3, pp. 457-511 | Numdam | MR | Zbl

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

[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., Volume 1 (1967), pp. 225-233 | MR | Zbl

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

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

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

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

[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, Volume 346 (2008) no. 7-8, pp. 395-400 | MR | Zbl

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

[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.), Volume 42, Math. Soc. Japan, Tokyo, 2004, pp. 203-207 | MR | Zbl

[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., Volume 149 (1998), pp. 1-8 | MR | Zbl

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

[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, pp. viii+206 | MR | Zbl

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

Cité par Sources :