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Levi-flat hypersurfaces immersed in complex surfaces of positive curvature. (Hypersurfaces Levi-plates immergées dans les surfaces complexes de courbure positive.) (French) Zbl 1070.37031

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 \(\mathcal{F}\) be a foliation by Riemann surfaces of class \(C^1\) on a compact manifold \(M\) of dimension \(3\). Assume that \(\mathcal{F}\) 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 \(\mathcal{F}\) is a quotient of the horizontal foliation of \({\mathbb{C}}P^1\times S^1\), or the image of \(\mathcal{F}\) is tangent to the subset where the curvature \(\Omega\) is zero.

MSC:

37F75 Dynamical aspects of holomorphic foliations and vector fields
32S65 Singularities of holomorphic vector fields and foliations
32C30 Integration on analytic sets and spaces, currents
32V40 Real submanifolds in complex manifolds
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