Generalizations of the $Q$-prime curvature via renormalized characteristic forms

TL;DR

This paper generalizes the $Q$-prime curvature on CR manifolds using renormalized characteristic forms, linking local invariants to global CR invariants and exploring related operators and transformation laws.

Contribution

It introduces a new family of $Q$-prime curvatures associated with renormalized characteristic forms, unifying local and global CR invariants and extending the theory of CR GJMS and $P$-prime operators.

Findings

The integral of the generalized $Q$-prime curvature matches Marugame's CR invariants.

Established transformation laws for the new curvatures under conformal changes.

Extended the theory of CR GJMS and $P$-prime operators to the generalized setting.

Abstract

The $Q$-prime curvature is a local pseudo-Einstein invariant on CR manifolds defined by Case and Yang, and Hirachi. Its integral, the total $Q$-prime curvature, gives a non-trivial global CR invariant. On the other hand, Marugame has constructed a family of global CR invariants via renormalized characteristic forms, which contains the total $Q$-prime curvature. In this paper, we introduce a generalization of the $Q$-prime curvature for each renormalized characteristic form, and show that its integral coincides with Marugame's CR invariant. We also study generalizations of the critical CR GJMS operator and the $P$-prime operator, which are related to the transformation laws of our new curvatures under conformal change.