Calculus of Variations and Geometric Measure Theory
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L. Martinazzi

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

created by martinazz on 06 Feb 2009
modified on 17 Jul 2018


Published Paper

Inserted: 6 feb 2009
Last Updated: 17 jul 2018

Journal: Math. Z.
Volume: 263
Pages: 307-329
Year: 2009

ArXiv: 0801.2729 PDF


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
\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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