Constant Q-curvature metrics near the Hyperbolic metric

Abstract

Let $(M,g)$ be a Poincar$\acute{\text{e}}$-Einstein manifold with a smooth defining function. In this note, we prove that there are infinitely many asymptotically hyperbolic metrics with constant $Q$-curvature in the conformal class of an asymptotically hyperbolic metric close enough to $g$. These metrics are parametrized by the elements in the kernel of the linearized operator of the prescribed constant $Q$-curvature equation. A similar analysis is applied to a class of fourth order equations arising in spectral theory.

arXiv comment29 pages. It is a correction of last version: A few words are added in Remark 1.1.to be more precise. Two typos in the reference and corrected. A missing weight in the statement of Corollary 2.5.is added. A sign is corrected in the sentence "When $n=4$"in the proof of Lemma 2.6