Hypersurfaces Levi-plates immergées dans les surfaces complexes de courbure positive

We show that there is no immersed compact Levi-flat hypersurface of class C1 in the complex projective plane, if the foliation by holomorphic curves carries a harmonic current which is absolutely continuous with respect to the Lebesgue measure, with a density bounded from above and below. This is a corollary of a rigidity result for immersed compact Levi-flat hypersurfaces in complex surfaces of non-negative curvature.