  We study renormalization-group flows by deforming a class of conformal
sigma-models. We consider overall scale factor perturbation of Einstein spaces
as well as more general anisotropic deformations of three-spheres. At leading
order in alpha, renormalization-group equations turn out to be Ricci flows. In
the three-sphere background, the latter is the Halphen system, which is exactly
solvable in terms of modular forms. We also analyze time-dependent deformations
of these systems supplemented with an extra time coordinate and time-dependent
dilaton. In some regimes time evolution is identified with
renormalization-group flow and time coordinate can appear as Liouville field.
The resulting space-time interpretation is that of a homogeneous isotropic
Friedmann-Robertson-Walker universe in axion-dilaton cosmology. We find as
general behaviour the superposition of a big-bang (polynomial) expansion with a
finite number of oscillations at early times. Any initial anisotropy disappears
during the evolution.
