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$Q$-prime curvature and scattering theory on strictly pseudoconvex domains

The $Q$-prime curvature is a local invariant of pseudo-Einstein contact forms on integrable strictly pseudoconvex CR manifolds. The transformation law of the $Q$-prime curvature under scaling is given in terms of a differential operator, called the $P$-prime operator, acting on the space of CR pluriharmonic functions. In this paper, we generalize these objects to the boundaries of asymptotically complex hyperbolic Einstein (ACHE) manifolds, which are partially integrable, strictly pseudoconvex CR manifolds, by using the scattering matrix for ACHE manifolds. In this setting, the $P$-prime operator is a self-adjoint pseudodifferential operator acting on smooth functions and the $Q$-prime curvature is globally determined by the ACHE manifold and the choice of a contact form on the boundary. We prove that the integral of the $Q$-prime curvature is a conformal primitive of the $Q$-curvature; in particular, it defines an invariant of ACHE manifolds whose boundaries admit a contact form with zero $Q$-curvature. We also apply the generalized $Q$-prime curvature to compute the renormalized volume of strictly pseudoconvex domains whose boundaries may not admit pseudo-Einstein structure.