Some generalizations of Halphen's equations
type: Item
instance of: scholarly article
title: Some generalizations of Halphen's equations
zbMATH Open document ID: 1251.32019
author: Adolfo Guillot
published in: Osaka Journal of Mathematics
publication date: 2012-02-17T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.ojm/1326291219
review text: The solutions of an ordinary differential equation in the complex domain may be multivalued. A semicomplete polynomial vector field on \({\mathbb C}^{n}\) is a polynomial vector field whose every solution is single-valued. The paper under review is devoted to the proof of the following theorem. Theorem. For every \(\rho\in{\mathbb C}\setminus\{0;1\}\), there exists a non empty open bounded subset \(B_{\rho}\subset{\mathbb C}\) such that the vector field \[ \begin{aligned} X_{\rho}(\alpha)&= [ z_{1}^{2}+z_{2}z_{3}(z_{2}-z_{3})(\{\alpha-\rho\}z_{2}+\rho\{1-\alpha\}z_{3})]\frac{\partial}{\partial z_{1}}\\ &+ z_{2}(z_{1}+\rho z_{2}z_{3}-\rho z_{3}^{2})\frac{\partial}{\partial z_{2}}+z_{3}(z_{1}+z_{2}^{2}-z_{2}z_{3})\frac{\partial}{\partial z_{3}} \end{aligned} \] is semicomplete for every \(\alpha\in\overline{B}_{\rho}\). A solution of any such \(X_{\rho}(\alpha)\) is defined in the complement of \(\infty\) in \({\mathbb CP}^{1}\) of an invariant component of the discontinuity domain of either a quasifuschian, a totally degenerate or a cusp Kleinian group (and all these situations appear). The proof is based on Lie theoretic methods connected with the so-called Halphen's equations. Finally an example is provided and some questions are addressed.
reviewed by: Salah Mehdi
Mathematics Subject Classification ID: 32M25
Mathematics Subject Classification ID: 30F40
zbMATH DE Number: 6006852
zbMATH Keywords: Halphen's equation
zbMATH Keywords: polynomial vector field
zbMATH Keywords: Kleinian group
MaRDI profile type: Publication
cites work: Two Theorems on Totally Degenerate Kleinian Groups
cites work: Q3994593
cites work: On boundaries of Teichmüller spaces and on Kleinian groups. I
cites work: Kleinian groups with small limit sets
cites work: Reductions of self-dual Yang-Mills fields and classical systems
cites work: First integrals of a generalized Darboux–Halphen system
cites work: Semicompleteness of homogeneous quadratic vector fields
cites work: On the equations of Halphen and \(\mathrm{SL}_{2}(\mathbb C)\)-actions
cites work: Semicomplete meromorphic vector fields on complex surfaces
cites work: Q3838089
cites work: On the algebraic non-integrability of the Halphen system
cites work: On boundaries of Teichmüller spaces and on Kleinian groups. II
cites work: Q4039828
cites work: Cusps are dense
cites work: Differential relations of theta functions
cites work: Q4824322
cites work: Singularités des flots holomorphes. (Singularities of holomorphic flows.)
cites work: Q3881471
cites work: Q4337982
description: scientific article; zbMATH DE number 6006852