Curvature restrictions for Levi-flat real hypersurfaces in complex projective planes

Adachi, Masanori; Brinkschulte, Judith

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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