An ambient metric framework for sharp compactness of the constant $Q_{2N}$-curvature problem

Abstract

We develop a geometric method for studying $C^{2N}$-compactness of positive solutions to the constant $Q_{2N}$-curvature equation on closed Riemannian $n$-manifolds, for every integer $N\ge1$ and every dimension $n>2N$. Assuming the GJMS operator is invertible, its Green function is positive, and the positive mass condition holds, we establish sharp dimension ranges for the compactness of the set of all positive solutions with prescribed positive constant $Q_{2N}$-curvature. The largest dimension for compactness is $24$ for $N=1$ ([21]), $2N+20$ for $N=2,3$ ([11]), $2N+19$ for $N=4,5$, $2N+18$ for $N=6,7,8$, $2N+17$ for $9\le N\le17$, and $2N+16$ for all $N\ge18$. The constructions in [4,5,33,11] for $N=1,2,3$ and in the companion paper [12] for $N\ge4$ give noncompact examples in every dimension exceeding the corresponding compactness bound. We show that these $L^{\infty}$-unbounded examples can be chosen to satisfy the same operator and mass hypotheses, proving sharpness within this class. Our method relates the Pohozaev quadratic forms to the Hessian $D^2\mathscr{F}(g_c)$, where $\mathscr{F}$ is the volume-normalized total $Q_{2N}$-curvature functional and $g_c$ is the standard round metric on $\mathbb{S}^n$. The ambient metric argument of [12] supplies an explicit formula for $D^2\mathscr{F}(g_c)$ on transverse-traceless tensors. The York decomposition projects metric perturbations onto this space. The scalar--vector--tensor decomposition represents the Pohozaev quadratic forms by explicit finite-dimensional matrices. We prove that these matrices are positive definite throughout the stated dimension range, a crucial step in establishing compactness.