Levi-flat hypersurfaces immerged in complex surfaces of positive curvature
type: Item
instance of: scholarly article
title: Levi-flat hypersurfaces immerged in complex surfaces of positive curvature
zbMATH Open document ID: 1042.53015
author: Bertrand Deroin
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2004-01-28T00:00:00Z
review text: A real hypersurface of a complex surface is said to be Levi-flat if it is foliated by holomorphic curves. It is known that there exists not compact Levi-flat hypersurface of class \(C^{2+\alpha}\) in the complex projective plane. The author shows that under an additional hypothesis on regularity there exists not the Levi-flat immersion either. This hypothesis is the existence of an absolutely continuous harmonic current (AC current) with respect to the Lebesgue measure, with a density bounded above and below by strictly positive constants. The following theorem is proved: Let \(S\) be a complex surface having a Hermitian metric of non negative Ricci curvature \(\Omega\), and \((M,\mathcal F)\) a foliation by Riemann surfaces of class \(C^1\) of a compact manifold of dimension 3. If \(\mathcal F\) possesses an AC harmonic current for all immersion \(\pi: M\to S\) of class \(C^1\) and holomorphic along the leaves then either \(\pi^\ast\Omega=0\), or \(\mathcal F\) possesses a foliated cycle \(C\) and \(\pi^\ast\Omega=0\) on the support of \(C\), or \(\mathcal F\) is a quotient of horizontal foliation of \(\mathbb CP^1\times\mathbb S^1\). This comes as a corollary of a rigidity result for Levi-flat immersions of class \(C^1\) of Riemann surface foliations having this regularity into complex surfaces of non negative Ricci curvature. Under the same hypotheses, there exists not an immersion into the projective complex plane of class \(C^1\), holomorphic along the leaves.
Mathematics Subject Classification ID: 53C12
Mathematics Subject Classification ID: 32S65
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 2034224
zbMATH Keywords: Levi-flat hypersurfaces
zbMATH Keywords: Levi-flat immersions
zbMATH Keywords: foliation by Riemann surfaces
zbMATH Keywords: Chern class
zbMATH Keywords: Hermitian metric of non negative Ricci curvature
zbMATH Keywords: absolutely continuous harmonic current
zbMATH Keywords: foliated cycle
zbMATH Keywords: rigidity
MaRDI profile type: Publication
full work available at URL: j.crma.2003.09.016
OpenAlex ID: W2015768271
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DOI: 10.1016/J.CRMA.2003.09.016
description: scientific article; zbMATH DE number 2034224