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

TL;DR

This paper classifies solutions to the higher order Liouville's equation on Euclidean space, linking analytic growth conditions to geometric properties like scalar curvature and conformal metrics, and characterizes when these metrics extend smoothly to spheres.

Contribution

It provides a complete classification of solutions with finite total Q-curvature, connecting analytic growth conditions to geometric and conformal properties.

Findings

Solutions characterized by growth rate and asymptotic behavior

Geometric characterization via scalar curvature at infinity

Smooth extension to the sphere occurs only for round metrics

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.