  We classify the solutions to the equation (- \Delta)^m u=(2m-1)!e^{2mu} on
R^{2m} giving rise to a metric g=e^{2u}g_{R^{2m}} with finite total
$Q$-curvature in terms of analytic and geometric properties. The analytic
conditions involve the growth rate of u and the asymptotic behaviour of \Delta
u(x) as |x|\to \infty. As a consequence we give a geometric characterization in
terms of the scalar curvature of the metric e^{2u}g_{R^{2m}} at infinity, and
we observe that the pull-back of this metric to $S^{2m}$ via the stereographic
projection can be extended to a smooth Riemannian metric if and only if it is
round.
