发表机构
Chinese University of Hong Kong; Hanyang University（香港中文大学; 汉阳大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文构造了球面上非局部共形平坦度量，使常数 $Q_{2N}$-曲率方程存在无界正解序列，证明所有阶的非紧性，并给出最优维度界。核心方法是通过 TT 方向 Hessian 的符号分析确定不定性。
AI 中文摘要
对于每个整数 $N\ge4$，我们在 $n$ 维单位球面 $\mathbb{S}^n$ 上构造一个固定的光滑、非局部共形平坦度量，使得常数 $Q_{2N}$-曲率方程允许一个 $L^\infty$-无界的正解序列。该构造适用于：当 $N=4,5$ 时 $n\ge2N+20$，当 $6\le N\le8$ 时 $n\ge2N+19$，当 $9\le N\le17$ 时 $n\ge2N+18$，当 $N\ge18$ 时 $n\ge2N+17$。对于每个 $L \in \mathbb{N} \cup \{0\}$，度量可以在 $C^L$ 范数下任意接近圆度量。结合已知的 $N=1,2,3$ 情形，这建立了所有阶的非紧性，且维度界我们预期是最优的。我们推导了在满足 $\operatorname{Ric}=(n-1)g$ 的闭爱因斯坦度量上，横向无迹（TT）方向的总 $Q_{2N}$-曲率的固定体积 Hessian 的显式公式。基于 Juhl 的公式，我们将该 Hessian 识别为 Lichnerowicz 拉普拉斯算子的 $N$ 次多项式，对每个 $N\in\mathbb{N}$ 和 $n>2N$ 成立。对于构造中由代数 Weyl 张量生成的度量扰动，固定气泡中心的二次约化能量是该 Hessian 限制在四维 TT 空间上的正倍数。对气泡尺度求导产生一个有限矩阵，其精确符号分析在所述维度范围内确立了所需的不定性。延拓到实数 $n$ 后，该矩阵在 $n_N=2N+a_*+c_*N^{-1}+O(N^{-2})$（当 $N\to\infty$）处从负定变为不定，其中 $a_*\approx16.201871$ 和 $c_*\approx13.512880$，解释了最终界 $n\ge2N+17$。
英文摘要
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 $n\ge2N+17$.
Comments52 pages, 1 figure
