Existence of Riemann surfaces through an equilibrium of a tangent holomorphic vector field

Abstract

It is shown that every holomorphic vector field that vanishes at a point where its derivative is invertible has a Riemann surface such that the vector field is tangent to the Riemann surface and such that the Riemann surface contains the point. In fact, we give a precise finite lower bound on the number of (germs of) such Riemann surfaces depending on the derivative of the vector field at such a point. We use for this a differentiable version of the Grobman-Hartman theorem. Additionally, we conjecture an upper bound on the number of independent integrals a vector field may have near an equilibrium. In the presentation, a newly developed notional system is used.