A CR proof for a global estimate of the Diederich--Fornaess index of Levi-flat real hypersurfaces

Yet another proof is given for a global estimate of the Diederich--Fornaess index of relatively compact domains with Levi-flat boundary, namely, the index must be smaller than or equal to the reciprocal of the dimension of the ambient space. This proof reveals that this kind of estimate makes sense and holds also for abstract compact Levi-flat CR manifolds.Comment: 7 pages, final version, to appear in "Complex Analysis and Geometry", Springer Proceedings in Mathematics & Statistic

A local expression of the Diederich--Fornaess exponent and the exponent of conformal harmonic measures

A local expression of the Diederich--Fornaess exponent of complements of Levi-flat real hypersurfaces is exhibited. This expression describes the correspondence between pseudoconvexity of their complements and positivity of their normal bundles, which was suggested in a work of Brunella, in a quantitative way. As an application, a connection between the Diederich--Fornaess exponent and the exponent of conformal harmonic measures is discussed.Comment: 11 pages, final version, copyright information adde

Weighted Bergman spaces of domains with Levi-flat boundary (Topology of pseudoconvex domains and analysis of reproducing kernels)

This note is an announcement of the author's recent works [1, 2, 3] on Levi-flat real hypersurfaces, which are motivated by the non-existence of smooth Levi-flat real hypersurface in the complex projective plane

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

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.Comment: 19 pages, final version, to appear in Annales de l'Institut Fourie