  On compact surfaces with or without boundary, Osgood, Phillips and Sarnak
proved that the maximum of the determinant of the Laplacian within a conformal
class of metrics with fixed area occurs at a metric of constant curvature and,
for negative Euler characteristic, exhibited a flow from a given metric to a
constant curvature metric along which the determinant increases. The aim of
this paper is to perform a similar analysis for the determinant of the
Laplacian on a non-compact surface whose ends are asymptotic to hyperbolic
funnels or cusps. In that context, we show that the Ricci flow converges to a
metric of constant curvature and that the determinant increases along this
flow.
