  As part of the general investigation of Ricci flow on complete surfaces with
finite total curvature, we study this flow for surfaces with asymptotically
conical (which includes as a special case asymptotically Euclidean) geometries.
After establishing long-time existence, and in particular the fact that the
flow preserves the asymptotically conic geometry, we prove that the solution
metric $g(t)$ expands at a locally uniform linear rate; moreover, the rescaled
family of metrics $t^{-1}g(t)$ exhibits a transition at infinite time inasmuch
as it converges locally uniformly to a complete, finite area hyperbolic metric
which is the unique uniformizing metric in the conformal class of the initial
metric $g_0$.
