AI 中文总结
研究在庞加莱 - 爱因斯坦流形共形无穷远处定义的分数阶GJMS算子\(P_{2\gamma}\)，推导了相关分数阶Yamabe常数的两个比较不等式，刻画等式蕴含刚性定理，为分数阶Yamabe常数单调性提供部分证据。
AI 中文摘要
本文主要关注在庞加莱 - 爱因斯坦流形的共形无穷远处定义的分数阶GJMS算子\(P_{2\gamma}\)。我们推导了与分数阶GJMS算子相关的分数阶 Yamabe 常数的两个比较不等式。一个是\(\gamma\in(1/2,1)\)时\(P_{1}\)与\(P_{2\gamma}\)之间的，另一个是\(\gamma\in(1,2)\)时\(P_{2}\)与\(P_{2\gamma}\)之间的。它们通过刻画等式蕴含刚性定理，与文献\(\cite{WZ1}\)的结果一起为分数阶 Yamabe 常数的单调性提供了部分证据。
英文摘要
In this paper, we mainly focus on the fractional GJMS operators $P_{2γ}$ which are defined on the conformal infinity of a Poincaré-Einstein manifold. We derive two comparison inequalities of the fractional Yamabe constants associated to the fractional GJMS operators. One is between $P_{1}$ and $P_{2γ}$ for $γ\in (1/2,1)$, and the other is between $P_{2}$ and $P_{2γ}$ for $γ\in (1,2)$. They both imply the rigidity theorems by characterizing the equalities. Together with the result in \cite{WZ1}, we partially provide some evidence for the monotonicity of the fractional Yamabe constants.
CommentsFractional GJMS operators
