Classification of solutions to the higher order Liouville's equation on $${\mathbb{R}^{2m}}$$

We classify the solutions to the equation (−Δ) m u=(2m−1)!e 2mu on $${\mathbb{R}^{2m}}$$ giving rise to a metric $${g=e^{2u}g_{\mathbb{R}^{2m}}}$$ 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 Δu at infinity. As a consequence we give a geometric characterization in terms of the scalar curvature of the metric $${e^{2u}g_{\mathbb{R}^{2m}}}$$ 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 roun