Variation of total Q-prime curvature on CR manifolds

TL;DR

This paper investigates how the total Q-prime curvature varies under domain deformations in CR manifolds, linking it to renormalized volume and CR invariant operators, thus advancing geometric analysis in CR geometry.

Contribution

It derives variational formulas for total Q-prime curvature and establishes its equivalence with renormalized volume, also analyzing associated CR invariant differential operators.

Findings

Total Q-prime curvature varies under domain deformation.

Total Q-prime curvature equals the renormalized volume.

Analysis of CR invariant differential operators of order 2(n+3).

Abstract

We derive variational formulas for the total Q-prime curvature under the deformation of strictly pseudoconvex domains in a complex manifold. We also show that the total Q-prime curvature agrees with the renormalized volume of such domains with respect to the complete Einstein-K\"ahler metric. In the appendix, by Rod Gover and the first author, we study the property of CR invariant differential operator of order 2(n+3) on 2n+1 dimensional CR manifolds that appears in the second variational formula of the total Q-prime curvature.