Complex structures adapted to smooth vector fields

Abstract

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.

Classification of complex polynomial vector fields in one complex variable

“…When it comes to proving existence of a quadratic differential with prescribed combinatorial and analytic properties (for instance, the so-called 'moduli problem' for quadradic differentials with closed trajectories [18]), the full classification of vector fields is excluded since these theorems are only applicable to a special subclass of vector fields, or they assume a finite area condition, which holomorphic vector fields with at least one zero never satisfy. Muciño-Raymundo [13] studies vector fields but also only covers a specific class.…”

Dynamics of singular complex analytic vector fields with essential singularities I

“…Summing up we get the following result (first introduced for (1)(2)(3)(4) in [50], [49], for (5) in the rational case [17], [18] and now expanded to cover (6) …”

“…Notation and conventions. The material covered in this section can be found, for the meromorphic category, in [66], [50], [49]. Our present contribution is to consider essential singularities.…”

“…Partial versions of the correspondence (1)-(6): J. Muciño-Raymundo et al [50], J. Muciño-Raymundo [49], A. Bustinduy et al [17], [18].…”

“…Flat metrics: K. Strebel [66], R. Peretz [57], J. Muciño-Raymundo et al [50], H. Masur et al [48], J. Muciño-Raymundo [49], M. Kontsevich et al [44], J. P. Bowman et al [15].…”