AI 中文总结
本文研究了六阶Q曲率在conformal度量中的正负性问题，发现当维度满足特定条件时，Q_g^{(6)}在某些情况下为正，而在其他情况下为负。
AI 中文摘要
给定n,m∈ℕ，使得n≥2m≥4，令g为R^n上的 conformally欧几里得度量，我们考虑当Q_g^{(2m)}非负且不恒为零时，下阶Q曲率Q_g^{(2k)}对于k∈{1,…,m−1}的正负性问题。此外，我们假设当n=2m时，度量g在无穷远处的标量曲率非负，或者当n>2m时，Q_g^{(2m)}满足慢衰减屏障条件。Gursky和Malchiodi在闭流形上有非负标量曲率的背景下，对于m=2和k=1得到了正的结果；Li和Xu以及Li, Wei和Xu在R^n上的conformally欧几里得度量背景下，对于m≥2和k∈{1,min(m−1,2)}也得到了正的结果。这些结果适用于所有n≥2m。考虑m≥4和k=3的情况，我们得到当n∈{2m,2m+1,…,4m−6}时，即对于这些维度，如果Q_g^{(2m)}≥0且Q_g^{(2m)}≠0，则Q_g^{(6)}>0。另一方面，在与Gursky和Malchiodi、Li和Xu以及Li, Wei和Xu的结果形成鲜明对比的情况下，我们发现当k=3且n≥N_m时，该问题的答案为负。在这种情况下，我们能够构造出conformally欧几里得度量的例子，使得Q_g^{(2m)}在处处正，但Q_g^{(6)}在某些点为负。通过立体投影，我们的例子扩展到与标准度量在S^n上 conformal的度量。
英文摘要
Given $n,m\in\mathbb{N}$ such that $n\ge2m\ge4$, letting $g$ be a conformally Euclidean metric on $\mathbb{R}^n$, we consider the question of positivity of the lower-order $Q$-curvatures $Q_g^{(2k)}$ for $k\in\left\{1,\dotsc,m-1\right\}$ when $Q_g^{(2m)}$ is assumed to be nonnegative and not identically zero. We assume moreover that the scalar curvature of the metric $g$ is nonnegative near infinity if $n=2m$ or that $Q_g^{(2m)}$ satisfies a slow decay barrier condition near infinity if $n>2m$. Positive results for this question have been obtained by Gursky and Malchiodi for $m=2$ and $k=1$ in the context of closed manifolds with nonnegative scalar curvature and by Li and Xu and Li, Wei, and Xu for $m\ge2$ and $k\in\left\{1,\min(m-1,2)\right\}$ in the context of conformally Euclidean metrics on $\mathbb{R}^n$. These results hold for all $n\ge2m$. Considering the case where $m\ge4$ and $k=3$, we obtain a positive result for this question when $n\in\left\{2m,2m+1,\dotsc,4m-6\right\}$, namely for these dimensions, we obtain that if $Q_g^{(2m)}\ge0$ and $Q_g^{(2m)}\not\equiv0$ in $\mathbb{R}^n$, then $Q_g^{(6)}>0$. On the other hand, in surprising contrast with the results of Gursky and Malchiodi, Li and Xu, and Li, Wei, and Xu, we find that the answer to this question is negative when $k=3$ and $n\ge N_m$ for some $N_m\in\mathbb{R}$. In this case, we are able to construct examples of conformally Euclidean metrics such that $Q_g^{(2m)}$ is positive everywhere, but $Q_g^{(6)}$ is negative at some point. By stereographic projection, our examples extend to metrics conformal to the standard metric on $\mathbb{S}^n$.
