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 - <p style="text-align: justify;">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 $0&lt;\gamma&lt;\frac{n}{2}$ for the conformal infinity $(M, [g])$. We generalise Guillarmou-Qing&#39;s positivity results in [8] to the higher order case. Namely, if $(X^{n+1}, g_+)$ $(n\geq 5)$ is a hyperbolic Poincar<span style="text-align: justify;">é</span>-Einstein manifold and there exists a smooth representative $g$ for the conformal infinity&nbsp; such that the scalar curvature $R_g$ is a positive constant and $Q_4$ is semi-positive on $(M, g)$,&nbsp; then $P_{2\gamma}$ is positive for $\gamma\in [1,2]$ and the first real scattering pole is less than $\frac{n}{2}-2$.</p>