Painlev\'e analysis of Ricci solitons over warped products

TL;DR

This paper applies Painlevé analysis to identify integrable cases of cohomogeneity one steady Ricci soliton equations, focusing on warped products and complex line bundles over Fano Kähler Einstein bases, revealing special dimensional cases.

Contribution

It introduces a Painlevé analysis approach to classify integrable Ricci soliton equations in specific geometric settings, highlighting unique dimensional cases.

Findings

Warped products with hypersurface dimension as a perfect square are integrable.

Two factors each of dimension 2 are singled out as integrable.

A 1-parameter family of solutions exists for complex line bundles in every even dimension.

Abstract

We carry out a Painlev\'e analysis to find the cases where the cohomogeneity one steady Ricci soliton equation can be integrable. We concentrate on two classes of solitons: warped products and complex line bundles over a Fano K\"ahler Einstein base. For warped products, the analysis singles out the case with one factor where the dimension of the hypersurface is a perfect square, with the $n=4$ particularly distinguished. The case with two factors each of dimension $2$ is also singled out by the analysis. In the case of complex line bundles, a 1-parameter family is singled out for every even dimension.