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

Abstract

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.

19 pages, final version, to appear in Annales de l'Institut Fourier

References in corpus (3)

• A global estimate for the Diederich--Fornaess index of weakly pseudoconvex domains
• A local expression of the Diederich--Fornaess exponent and the exponent of conformal harmonic measures
• The Diederich-Fornæss exponent and non-existence of Stein domains with Levi-flat boundaries

Cited by in corpus (5)

• Diederich-Fornaess and Steinness indices for abstract CR manifolds
• Local criteria for non embeddability of Levi-flat manifolds
• On a hyperconvex manifold without non-constant bounded holomorphic functions
• On the normal bundle of Levi-flat real hypersurfaces
• A CR proof for a global estimate of the Diederich--Fornaess index of Levi-flat real hypersurfaces