Levi-flat hypersurfaces immerged in complex surfaces of positive curvature
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.
Hermitian metric of non negative Ricci curvature
scientific article; zbMATH DE number 2034224