Levi-flat hypersurfaces immerged in complex surfaces of positive curvature

We prove that there is no Levi-flat immersion of class <f>C1</f> of a Riemann surface foliation of class <f>C1</f> of a <f>3</f>-dimensional compact manifold in the complex projective plane, if the foliation carries a harmonic current which is absolutely continuous with respect to Lebesgue measure, with a density bounded from above and below. This comes as a corollary of a rigidity result for Levi-flat immersions of class <f>C1</f> of Riemann surface foliations having this regularity into complex surfaces of non negative Ricci curvature. To cite this article: B. Deroin, C. R. Acad. Sci. Paris, Ser. I 337 (2003). [Copyright &y& Elsevier]