Constant Q-curvature metrics near the Hyperbolic metric

TL;DR

This paper demonstrates the existence of infinitely many constant Q-curvature metrics near a given hyperbolic metric on Poincaré-Einstein manifolds, using linearized operator analysis and spectral theory techniques.

Contribution

It establishes the abundance of constant Q-curvature metrics close to a hyperbolic metric, parametrized by the kernel of the linearized operator, extending previous results in conformal geometry.

Findings

Infinitely many constant Q-curvature metrics exist near a hyperbolic metric.

Metrics are parametrized by the kernel of the linearized operator.

Analysis applies to related fourth order spectral equations.

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.