Communications in Analysis and Geometry
Volume 12 (2004)
Painlevé Expansions, Cohomogeneity One Metrics and Exceptional Holonomy
Pages: 887 – 926
In this paper we continue our study ([DW1] - [DW4]) of ordinary differential equations arising as reductions of the Einstein equations. One way of performing this reduction is to require the Einstein metrics to be of cohomogeneity one -- that is, they are required to be invariant under the action of a group with principal orbits of codimension one.
In [DW4] we introduced variables so that the cohomogeneity one Ricci- flat equations became a constrained flow of an ODE system with quadratic nonlinearities. This enabled us to perform a Painlevé-Kowalewski analysis (cf. [ARS], [AvM]) of these equations in the case when the isotropy representation had two inequivalent summands. In Painlevé-Kowalewski analysis, one looks for singular solutions of a system of ODEs (containing several parameters) given by Painlevé expansions, i.e., meromorphic expansions in (a rational power of) the independent variable. The singularity (poles or branch points) is movable (i.e., its position can be continuously varied), and should be distinguished (in the particular case of the Einstein equations) from singularities of the metric. The general philosophy of this method is that the existence of large families of Painlevé expansions should be associated with "nice" properties of the equations, and in general occurs for only special values of the parameters in the equations. In particular, if for each dependent variable in the equations there is a corresponding family of Painlevé expansions which depends on the maximum number of degrees of freedom and in which that dependent variable actually blows up, then this is regarded as a strong indication that the equations are "integrable".
In this paper we shall prove some general results about the Painlevé analysis of the cohomogeneity 1 Ricci-flat equations in the situation when the isotropy representation of the principal orbit consists of pairwise inequivalent representations. We then apply these results to examples of more complicated orbit types than those in [DW4], including some which have recently become relevant in the study of metrics of exceptional holonomy in string theory ([CGLP1], [BGGG]).