In this paper we study the flat geometry and real dynamics of meromorphic vector fields on compact Riemann surfaces. Necessary and sufficient conditions to assert the existence of meromorphic vector fields with prescribed singularities are given. A characterization of the real dynamics of meromorphic vector fields is also given. Several explicit examples of meromorphic vector fields, using singular flat metrics, are provided.

Abstract. We tackle the problem of understanding the geometry and dynamics of singular complex analytic vector fields X with essential singularities on a Riemann surface M (compact or not). Two basic techniques are used. (a) In the complex analytic category on M , we exploit the correspondence between singular vector fields X, differential forms ω X (with ω X (X) ≡ 1), orientable quadratic differentials ω X ⊗ ω X , global distinguished parameters Ψ X (z) = z ω X , and the Riemann surfaces R X of the above parameters. (b) We use the fact that all singular complex analytic vector fields can be expressed as the global pullback via certain maps of the holomorphic vector fields on the Riemann sphere, in particular, via their respective Ψ X .We show that under certain analytical conditions on Ψ X , the germ of a singular complex analytic vector field determines a decomposition in angular sectors; center C, hyperbolic H, elliptic E, parabolic P sectors but with the addition of suitable copies of a new type of entire angular sector E , stemming from X(z) = e z ∂ ∂z . This extends the classical theorems of A. A. Andronov et al. on the decomposition in angular sectors of real analytic vector field germs.The Poincaré-Hopf index theory for Re (X) local and global on compact Riemann surfaces, is extended so as to include the case of suitable isolated essential singularities.The inverse problem: determining which cyclic words W X , comprised of hyperbolic, elliptic, parabolic and entire angular sectors, it is possible to obtain from germs of singular analytic vector fields, is also answered in terms of local analytical invariants.We also study the problem of when and how a germ of a singular complex analytic vector field having an essential singularity (not necessarily isolated) can be extended to a suitable compact Riemann surface.Considering the family of entire vector fields E(d) = {X(z) = λe P (z) ∂ ∂z } on the Riemann sphere, where P (z) is a polynomial of degree d and λ ∈ C * , we completely characterize the local and global dynamics of this class of vector fields, compute analytic normal forms for d = 1, 2, 3, and show that for d ≥ 3 there are an infinite number of topological (phase portrait) classes of Re(X), for X ∈ E(d). These results are based on the work of R. Nevanlinna, A. Speisser and M. Taniguchi on entire functions Ψ X .Finally, on the topological decomposition of real vector fields into canonical regions, we extend the results of L. Markus and H. E. Benzinger to meromorphic on C vector fields X, with an essential singularity at ∞ ∈ C, whose Ψ −1 X have d logarithmic branch points over d finite asymptotic values and d logarithmic branch points over ∞.

Let {Xg} be a family of rotated singular real foliations in the Riemann sphere which is the result of the rotation of a meromorphic vector field X with zeros and poles of multiplicity one. We prove that the set of bifurcation values, in the circle {8}, is for each family a set with at most a finite number of accumulation points. A condition which implies a finite number of bifurcation values is given. We also show that the property of having an infinite set of bifurcation values defines open but not dense sets in the space of meromorphic vector fields with fixed degree.

We study vector fields on the plane having only isochronous centres. The most familiar examples are isochronous vector fields, they are the real parts of complex polynomial vector fields on C having all their zeroes of centre type. We describe the number NðsÞ of topologically inequivalent isochronous (singular) foliations that can appear for degree s, up to orientation preserving homeomorphisms. For each s, there exists a real analytic variety I ðsÞ parametrizing the isochronous vector fields of degree s, the group of complex automorphisms of the plane AutðCÞ acts on it. Furthermore, if 2 # s # 7, then I ðsÞ is a non-singular real analytic variety of dimension s þ 3, and their number of connected components is bounded by 2NðsÞ. An explicit formula for the residues of the rational 1-form, canonically associated with a complex polynomial vector field with simple zeroes, is given. A collection of residues (i.e. periods) does not characterize an isochronous vector field, even up to complex automorphisms of C. An exact bound for the number of isochronous vector fields, up to AutðCÞ, having the same collection of residues (periods) is given. We develop several descriptions of the quotient space I ðsÞ=AutðCÞ using residues, weighted s-trees and singular flat Riemannian metrics associated with isochronous vector fields.

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