An ambient metric framework for sharp compactness of the constant Q2N-curvature problem
Abstract
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.
Related
- The asymptotic Plateau problem for Hypersurfaces of constant Hk curvature in hyperbolic space
- Hyperbolic Completion of Newton's Off-Center Orbit Problem: Constant-Curvature Rigidity, SO(2,1) Symmetry, and Magnetic Uniqueness
- Noncompactness for the constant Q2N-curvature problem
- A remark on the compactness of manifolds with non-negative Ricci curvature and uniformly convex boundary
- The CR Nirenberg Problem, Part I: Blow-up Analysis and Compactness
BibTeX
@misc{gong2026ambient,
title = {{An ambient metric framework for sharp compactness of the constant $Q_2N$-curvature problem}},
author = {Liuwei Gong and Seunghyeok Kim and Juncheng Wei},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06551},
eprint = {2611.06551},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06551},
}
RIS
TY - UNPB
TI - An ambient metric framework for sharp compactness of the constant Q_2N-curvature problem
AU - Liuwei Gong
AU - Seunghyeok Kim
AU - Juncheng Wei
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.06551
UR - https://arxiv.org/abs/2611.06551
AB - 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≥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 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≤ 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 D^2ℱ(g_c), where ℱ is the volume-normalized total Q_2N-curvature functional and g_c is the standard round metric on 𝕊^n. The ambient metric argument of [12] supplies an explicit formula for D^2ℱ(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.
ER -