标题： 高阶常$Q$-曲率共形度量的紧性 显示英文标题

标题： Compactness of conformal metrics with constant $Q$-curvature of higher order

摘要： Let $k\ge1$ be a positive integer and let $P_g$ be the GJMS operator $P_{g}$ of order $2k$ on a closed Riemannian manifold $(M,g)$ of dimension $n>2k$. 我们研究了具有指定常正$Q$曲率的$g$的共形度量的紧性，其阶数为$2k$，或者等价地，研究了$2k$阶$Q$曲率方程的正解集的紧性。 在对$P_{g}$的自然保正性条件下，我们建立了紧性，对于任意的$1 \le k < \frac{n}{2}$，在以下假设下：$(M,g)$是局部共形平坦的，且$P_g$在$M$中具有正质量，或$2k+1 \le n \le 2k+5$和$P_g$在$M$中具有正质量，或$n \ge 2k+4$和$|\text{W}_g|_g >0$在$M$中。 对于任意的$1 \le k < \frac{n}{2}$，$P_g$的表达式不是显式的，这阻碍了紧性证明。我们通过依赖 Juhl 著名的$P_g$的递归公式，对$Q$曲率方程的解进行细致的爆破分析，并证明了$P_g$的 Weyl 消失结果。这是针对任意的$1 \le k < \frac{n}{2}$的第一个紧性结果，也是首次成功利用 Juhl 的公式来获得紧性的实例。 我们的结果还暗示，对于$2k$阶$Q$曲率方程紧性临界维数随着$k \to + \infty$而发散。 显示英文摘要

摘要： Let $k\ge1$ be a positive integer and let $P_g$ be the GJMS operator $P_{g}$ of order $2k$ on a closed Riemannian manifold $(M,g)$ of dimension $n>2k$. We investigate the compactness of the set of conformal metrics to $g$ with prescribed constant positive $Q$-curvature of order $2k$- or, equivalently, of the set of positive solutions for the $2k$-th order $Q$-curvature equation. Under a natural positivity-preserving condition on $P_{g}$ we establish compactness, for an arbitrary $1 \le k < \frac{n}{2}$, under the following assumptions: $(M,g)$ is locally conformally flat and $P_g$ has positive mass in $M$, or $2k+1 \le n \le 2k+5$ and $P_g$ has positive mass in $M$, or $n \ge 2k+4$ and $|\text{W}_g|_g >0$ in $M$. For an arbitrary $1 \le k < \frac{n}{2}$, the expression of $P_g$ is not explicit, which is an obstacle to proving compactness. We overcome this by relying on Juhl's celebrated recursive formulae for $P_g$ to perform a refined blow-up analysis for solutions of the $Q$-curvature equation and to prove a Weyl vanishing result for $P_g$. This is the first compactness result for an arbitrary $1 \le k < \frac{n}{2}$ and the first successful instance where Juhl's formulae are used to yield compactness. Our result also hints that the threshold dimension for compactness for the $2k$-th order $Q$-curvature equation diverges as $k \to + \infty$.