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 round.

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