An ambient metric framework for sharp compactness of the constant Q2N-curvature problem

We develop a geometric method for studying C2N-compactness of positive solutions to the constant Q2N-curvature equation on closed Riemannian n-manifolds, for every integer N≥1 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 Q2N-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≤ N≤17, and $2N+16$ for all N≥18. The constructions in [4,5,33,11] for $N=1,2,3$ and in the companion paper [12] for N≥4 give noncompact examples in every dimension exceeding the corresponding compactness bound. We show that these L∞-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 D²F(gc), where F is the volume-normalized total Q2N-curvature functional and gc is the standard round metric on Sⁿ. The ambient metric argument of [12] supplies an explicit formula for D²F(gc) 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.