*Published Paper*

**Inserted:** 6 feb 2009

**Last Updated:** 17 jul 2018

**Journal:** Math. Z.

**Volume:** 263

**Pages:** 307-329

**Year:** 2009

**Abstract:**

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.

**Keywords:**
Q-curvature, Paneitz operator

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