On the scattering operators for ACHE metrics of Bergman type on strictly pseudoconvex domains

TL;DR

This paper studies the positivity and spectral properties of scattering operators related to ACHE metrics on strictly pseudoconvex domains, extending known results from the real case to the CR setting.

Contribution

It demonstrates positivity and maximum principles for fractional scattering operators and curvature on CR manifolds with positive Webster scalar curvature, extending real case results.

Findings

Positive spectrum for $P_{2eta}$ when boundary Webster scalar curvature is positive

Fractional curvature $Q_{2eta}$ is positive under the same conditions

Provides energy extension formulas for the scattering operators

Abstract

The scattering operators associated to an ACHE metric of Bergman type on a strictly pseudovonvex domain are a one-parameter family of CR-conformally invariant pseudodifferntial operators of Heisenberg class with respect to the induced CR structure on the boundary. In this paper, we mainly show that if the boundary Webster scalar curvature is positive, then for $\gamma\in(0,1)$ the renormalised scattering operator $P_{2\gamma}$ has positive spectrum and satisfies the maximum principal; moreover, the fractional curvature $Q_{2\gamma}$ is also positive. This is parallel to the result of Guillarmou-Qing \cite{GQ} for the real case. We also give two energy extension formulae for $P_{2\gamma}$, which are parallel to the energy extension given by Chang-Case \cite{CC} for the real case.