Uniformizing complex ODEs and applications

TL;DR

This paper develops a method to estimate the domain size of solutions to meromorphic vector fields near their poles, with applications to polynomial vector fields and classical equations like Halphen's, linking complex analysis and geometry.

Contribution

Introduces a quantitative approach to analyze the domain of solutions of meromorphic vector fields and applies it to polynomial vector fields and classical equations.

Findings

Provides a confinement theorem for solutions of polynomial vector fields on ^n.

Identifies obstructions to realizing certain vector fields as global holomorphic fields on Ke4hler manifolds.

Proposes a new approach to classical equations, exemplified with Halphen equations.

Abstract

We introduce a method to estimate the size of the domain of definition of the solutions of a meromorphic vector field on a neighborhood of its pole divisor. The corresponding techniques are, in a certain sense, quantitative versions of some well-known phenomena related to the presence of metrics with positive curvature. Several applications of these ideas are provided including a type of "confinement theorem" for solutions of complete polynomial vector fields on $\mathbb{C}^n$ and obstructions for certain (germs of) vector fields to be realized by a global holomorphic vector field on a compact K\"ahler manifold. As a complement a new approach to certain classical equations is proposed and detailed in the case of Halphen equations.