ON MINIMAL SURFACES WHOSE GAUSS MAP IS INVARIANT BY A HOLOMORPHIC FOLIATION

We study real minimal surfaces : M^{2 4} under the hypothesisthat the holomorphic Gauss map of the immersion is invariantby a holomorphic foliation with singularities. We give a sortof Huber–Osserman theorem regarding the algebraicity ofthe holomorphic Gauss map.     

On the existence of stable compact leaves for transversely holomorphic foliations

We prove that a transversely holomorphic foliation on a compact manifold exhibits some compact leaf with finite holonomy group, provided that the set of compact leaves is not a zero measure set. A similar result is stated for groups of complex diffeomorphisms and periodic orbits.     

Holomorphic vector fields transverse to polydiscs

In this article, we study holomorphic vector fields transverseto the boundary of a polydisc in ⁿ, n 3. We prove that, undera suitable hypothesis of transversality with the boundary ofthe polydisc, the foliation is the pull-back of a linear hyperbolicfoliation via a locally injective holomorphic map. This is then 3 version for one-dimensional foliations of a previous resultproved for n = 2 by Brunella and Sad and for codimension-onefoliations by Ito and Scárdua.     

Complex polynomial vector fields having an orbit with finite total curvature

In this paper we address the following questions: (i) Let \({C \subset \mathbb{C}^2}\) be an orbit of a polynomial vector field which has finite total Gaussian curvature. Is C contained in an algebraic curve? (ii) What can be said of a polynomial vector field which has a finitely curved transcendent orbit? We give a positive answer to (i) under some non-degeneracy conditions on the singularities of the projective foliation induced by the vector field. For vector fields with a slightly more general class of singularities we prove a classification result that captures rational pull-backs of Poincaré-Dulac normal forms.