Existence results for the higher-order $Q$-curvature equation

TL;DR

This paper establishes existence results for higher-order Q-curvature equations on closed Riemannian manifolds under specific geometric conditions, extending the understanding of conformal geometry and curvature prescription problems.

Contribution

It provides new existence theorems for the $Q$-curvature equation of order $2k$, under positivity assumptions on the Yamabe invariant and Green's function, including cases with positive mass and non-locally conformally flat manifolds.

Findings

Existence results hold when the Yamabe invariant is positive.

Positivity of the Green's function and mass are key conditions.

Results depend on the dimension and conformal flatness of the manifold.

Abstract

We obtain existence results for the $Q$-curvature equation of order $2k$ on a closed Riemannian manifold of dimension $n\ge 2k+1$, where $k\ge1$ is an integer. We obtain these results under the assumptions that the Yamabe invariant of order $2k$ is positive and the Green's function of the corresponding operator is positive, which are satisfied for instance when the manifold is Einstein with positive scalar curvature. In the case where $2k+1\le n\le2k+3$ or $(M,g)$ is locally conformally flat, we assume moreover that the operator has positive mass. In the case where $n\ge2k+4$ and $(M,g)$ is not locally conformally flat, the results essentially reduce to the determination of the sign of a complicated constant depending only on $n$ and $k$.