On the positivity of scattering operators for Poincar\'{e}-Einstein manifolds

TL;DR

This paper investigates the positivity of scattering operators on Poincaré-Einstein manifolds, establishing conditions under which these operators are positive and analyzing implications for scattering poles.

Contribution

It proves the positivity of fractional GJMS operators for certain Poincaré-Einstein manifolds under specific geometric conditions, extending understanding of scattering operators.

Findings

Positivity of $P_{2 ext{"}gamma}$ for $ ext{"}gamma$ in (1,2).

First scattering pole is less than $rac{n}{2}-2$.

Conditions include local conformal flatness, positive scalar curvature, and $Q_4>0$.

Abstract

In this paper, we mainly study the scattering operators for the Poincar\'{e}-Einstein manifolds. Those operators give the fractional GJMS operators $P_{2\gamma}$ for the conformal infinity. If a Poincar\'{e}-Einstein manifolds $(X^{n+1}, g_+)$ is locally conformally flat and there exists an representative $g$ for the conformal infinity $(M, [g])$ such that the scalar curvature $R$ is a positive constant and $Q_4>0$, then we prove that $P_{2\gamma}$ is positive for $\gamma\in (1,2)$ and thus the first real scattering pole is less than $\frac{n}{2}-2$.