  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.
