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Zbl 1070.37031
Deroin, Bertrand
Levi-flat hypersurfaces immersed in complex surfaces of positive curvature. (Hypersurfaces Levi-plates immergées dans les surfaces complexes de courbure positive.) (French)
[J] Ann. Sci. Éc. Norm. Supér. (4) 38, No. 1, 57-75 (2005). ISSN 0012-9593

The author presents a dynamical approach to Levi-flat hypersurfaces in a complex surface, obtained by foliation by holomorphic curves. The main result is Theorem 1.2. Let $S$ be a complex surface for which the anti-canonical bundle $K_S$ has a metric of class $C^2$ of curvature $\Omega$ positive or null. Let $\cal{F}$ be a foliation by Riemann surfaces of class $C^1$ on a compact manifold $M$ of dimension $3$. Assume that $\cal{F}$ has a harmonic current absolutely continuous with respect to the Lebesgue measure with a density bounded from above and from below by two positive constants. Then there exists a $C^1$-immersion $L:M\to S$, holomorphic along the leaves, such that either $\cal{F}$ is a quotient of the horizontal foliation of ${\Bbb{C}}P^1\times S^1$, or the image of $\cal{F}$ is tangent to the subset where the curvature $\Omega$ is zero.
[Vasile Oproiu (Iaşi)]

MSC 2000:: *37F75 Holomorphic foliations and vector fields
32S65 Singularities of holomorphic vector fields
32C30 Integration on analytic sets and spaces
32V40 Real submanifolds in complex manifolds

Keywords: CR submanifolds; Levi form; foliation by holomorphic curves

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