Uniqueness of conformal metrics with constant Q-curvature on closed Einstein manifolds

TL;DR

This paper proves that on certain closed Einstein manifolds with positive scalar curvature, the only conformal metrics with constant Q-curvature are scalar multiples of the original metric, establishing a uniqueness result.

Contribution

It establishes a uniqueness theorem for conformal metrics with constant Q-curvature on closed Einstein manifolds not conformally equivalent to the sphere.

Findings

Uniqueness of conformal metrics with constant Q-curvature on specified manifolds

No other conformal metrics with constant Q-curvature exist besides scalar multiples of the original

Results apply to manifolds with positive scalar curvature not conformally diffeomorphic to the sphere

Abstract

On a smooth, closed Riemannian manifold $(M,g)$ of dimension $n\ge3$ with positive scalar curvature and not conformally diffeomorphic to the standard sphere, we prove that the only conformal metrics to $g$ with constant Q-curvature of order 4 are the metrics $\lambda g$ with $\lambda>0$ constant.