[Levi-flat hypersurfaces immerged in complex surfaces of positive curvature]
We prove that there is no Levi-flat immersion of class C1 of a Riemann surface foliation of class C1 of a 3-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 C1 of Riemann surface foliations having this regularity into complex surfaces of non negative Ricci curvature.
Nous démontrons qu'il n'y a pas d'immersion Levi-plate de classe C1 d'un feuilletage par surfaces de Riemann de classe C1 d'une variété compacte de dimension 3 dans le plan projectif complexe, si le feuilletage possède un courant harmonique absolument continu par rapport à la mesure de Lebesgue, avec une densité bornée supérieurement et inférieurement. Ceci découle d'un résultat de rigidité pour les immersions Levi-plates d'un feuilletage ayant la même régularité, à valeurs dans une surface complexe de courbure de Ricci positive ou nulle.
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Bertrand Deroin 1
@article{CRMATH_2003__337_12_777_0, author = {Bertrand Deroin}, title = {Hypersurfaces {Levi-plates} immerg\'ees dans les surfaces complexes de courbure positive}, journal = {Comptes Rendus. Math\'ematique}, pages = {777--780}, publisher = {Elsevier}, volume = {337}, number = {12}, year = {2003}, doi = {10.1016/j.crma.2003.09.016}, language = {fr}, }
Bertrand Deroin. Hypersurfaces Levi-plates immergées dans les surfaces complexes de courbure positive. Comptes Rendus. Mathématique, Volume 337 (2003) no. 12, pp. 777-780. doi : 10.1016/j.crma.2003.09.016. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2003.09.016/
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