Symmetries of complex analytic vector fields with an essential singularity on the Riemann sphere

Abstract

We consider the family $\mathcal{E}(s,r,d)=\{X(z)=\frac{Q(z)}{P(z)}{e}^{E(z)}\frac{\partial }{\partial z}\},$ with $Q,P,E$ polynomials, deg$Q=s$, deg$(P)=r$ and deg$(E)=d$, of singular complex analytic vector fields $X$ on the Riemann sphere $\hat{\mathbb{C}}$. For $d\ge 1$, $X\in \mathcal{E}(s,r,d)$ has $s$ zeros and $r$ poles on the complex plane and an essential singularity at infinity. Using the pullback action of the affine group $Aut(\mathbb{C})$ and the divisors for $X$, we calculate the isotropy groups $Aut(\mathbb{C}{)}_X$ and the discrete symmetries for $X\in \mathcal{E}(s,r,d)$. Each subfamily $\mathcal{E}(s,r,d{)}_{id}$, of those $X$ with trivial isotropy group in $Aut(\mathbb{C})$, is endowed with a holomorphic trivial principal $Aut(\mathbb{C})$-bundle structure. Necessary and sufficient conditions in order to ensure the equality $\mathcal{E}(s,r,d)=\mathcal{E}(s,r,d{)}_{id}$ and those $X\in \mathcal{E}(s,r,d)$ with non-trivial isotropy are realized. Explicit global normal forms for $X\in \mathcal{E}(s,r,d)$ are presented. A natural dictionary between vector fields, 1-forms, quadratic differentials and functions is extended to include the presence of non-trivial discrete symmetries $\Gamma <Aut(\mathbb{C})$.

arXiv comment25 pages, 5 figures

Cited by in corpus (1)

• On the geometry, flows and visualization of singular complex analytic vector fields on Riemann surfaces