On constant Q-curvature metrics with isolated singularities

Abstract

In this paper we derive a refined asymptotic expansion, near an isolated singularity, for conformally flat metrics with constant positive Q-curvature and positive scalar curvature. The condition that the metric has constant Q-curvature forces the conformal factor to satisfy a fourth order nonlinear partial differential equation with critical Sobolev growth, whose leading term is the bilaplacian. We model our results on a similar asymptotic expansion for conformally flat, constant scalar curvature metrics proven by Korevaar, Mazzeo, Pacard, and Schoen. Along the way we analyze the linearization of the Q-curvature equation about the Delaunay metrics recently discovered by Frank and König, which may be of independent interest.

arXiv comment34 pages, 1 figure, long overdue

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• Classification of positive solutions to a nonlinear biharmonic equation with critical exponent
• Asymptotic symmetry and local behavior of solutions of higher order conformally invariant equations with isolated singularities
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Cited by in corpus (4)

• Constant Q-curvature metrics with Delaunay ends: the nondegenerate case
• On a fourth order conformal invariant
• Local estimates for conformal $Q$-curvature equations
• Local behavior of positive solutions of higher order conformally invariant equations with a singular set