Noncompactness for the constant $Q_{2N}$-curvature problem

Abstract

For every integer $N\ge4$, we construct a fixed smooth, non-locally-conformally-flat metric on the $n$-dimensional unit sphere $\mathbb{S}^n$ for which the constant $Q_{2N}$-curvature equation admits an $L^\infty$-unbounded sequence of positive solutions. The construction applies for $n\ge2N+20$ when $N=4,5$, $n\ge2N+19$ when $6\le N\le8$, $n\ge2N+18$ when $9\le N\le17$, and $n\ge2N+17$ when $N\ge18$. For each $L \in \mathbb{N} \cup \{0\}$, the metric can be chosen arbitrarily close to the round metric in the $C^L$ norm. Together with the known cases $N=1,2,3$, this establishes noncompactness at every order, with dimension bounds that we expect to be optimal. We derive an explicit formula for the fixed-volume Hessian of total $Q_{2N}$-curvature in transverse-traceless (TT) directions at closed Einstein metrics satisfying $\operatorname{Ric}=(n-1)g$. Building on Juhl's formulas, we identify this Hessian, for every $N\in\mathbb{N}$ and $n>2N$, as a degree-$N$ polynomial in the Lichnerowicz Laplacian. For the metric perturbations generated by algebraic Weyl tensors in the construction, the quadratic reduced energy with fixed bubble center is a positive multiple of this Hessian restricted to a four-dimensional TT space. Differentiation with respect to the bubble scale then yields a finite matrix whose exact sign analysis establishes the required indefiniteness in the stated dimension ranges. After continuation to real $n$, this matrix changes from negative definite to indefinite at $n_N=2N+a_*+c_*N^{-1}+O(N^{-2})$ as $N\to\infty$, with $a_*\approx16.201871$ and $c_*\approx13.512880$, explaining the eventual bound $n\ge2N+17$.