Bianchi-IX, Darboux-Halphen and Chazy-Ramanujan

TL;DR

This paper investigates the connections between Bianchi-IX metrics, Darboux-Halphen systems, and Chazy-Ramanujan equations, exploring their integrability and implications for gravitational instantons and Ricci flow.

Contribution

It provides a detailed analysis of the relationship between self-duality in Bianchi-IX metrics and integrability of related differential systems.

Findings

Self-duality conditions relate to integrability of the Darboux-Halphen system.

Certain solutions exhibit Ricci flow behavior.

A list of integrable and near-integrable systems connected to Bianchi IX geometry.

Abstract

Bianchi-IX four metrics are $SU(2)$ invariant solutions of vacuum Einstein equation, for which the connection-wise self-dual case describes the Euler Top, while the curvature-wise self-dual case yields the Ricci flat classical Darboux-Halphen system. It is possible to see such a solution exhibiting Ricci flow. The classical Darboux-Halphen system is a special case of the generalized one that arises from a reduction of the self-dual Yang-Mills equation and the solutions to the related homogeneous quadratic differential equations provide the desired metric. A few integrable and near-integrable dynamical systems related to the Darboux-Halphen system and occurring in the study of Bianchi IX gravitational instanton have been listed as well. We explore in details whether self-duality implies integrability.