TY - JOUR T1 - On the Positivity of Scattering Operators for Poincaré-Einstein Manifolds AU - Wang , Fang JO - Journal of Mathematical Study VL - 2 SP - 186 EP - 199 PY - 2021 DA - 2021/02 SN - 54 DO - http://doi.org/10.4208/jms.v54n2.21.05 UR - https://global-sci.org/intro/article_detail/jms/18616.html KW - Scattering operators, fractional GJMS, positivity, Poincaré-Einstein. AB -

In this paper, we mainly study the scattering operators for a Poincaré-Einstein manifold $(X^{n+1}, g_+)$, which define the fractional GJMS operators $P_{2\gamma}$ of order $2\gamma$ for $0<\gamma<\frac{n}{2}$ for the conformal infinity $(M, [g])$. We generalise Guillarmou-Qing's positivity results in [8] to the higher order case. Namely, if $(X^{n+1}, g_+)$ $(n\geq 5)$ is a hyperbolic Poincaré-Einstein manifold and there exists a smooth representative $g$ for the conformal infinity  such that the scalar curvature $R_g$ is a positive constant and $Q_4$ is semi-positive on $(M, g)$,  then $P_{2\gamma}$ is positive for $\gamma\in [1,2]$ and the first real scattering pole is less than $\frac{n}{2}-2$.