#### Vol. 309, No. 2, 2020

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Functional determinant on pseudo-Einstein 3-manifolds

### Ali Maalaoui

Vol. 309 (2020), No. 2, 421–436
##### Abstract

Given a three-dimensional pseudo-Einstein CR manifold $\left(M,{T}^{1,0}M,𝜃\right)$, we establish an expression for the difference of determinants of the Paneitz type operators ${A}_{𝜃}$, related to the problem of prescribing the ${Q}^{\prime }$-curvature, under the conformal change $𝜃↦{e}^{w}𝜃$ with $w\in \mathsc{𝒫}$ the space of pluriharmonic functions. This generalizes the expression of the functional determinant in four-dimensional Riemannian manifolds established in (Proc. Amer. Math. Soc. 113:3 (1991), 669–682). We also provide an existence result of maximizers for the scaling invariant functional determinant as in (Ann. of Math.$\left(2\right)$ 142:1 (1995), 171–212).

##### Keywords
pseudo-Einstein CR manifolds, functional determinant, the $P'$-operator
##### Mathematical Subject Classification
Primary: 58J35, 58J50
Secondary: 32V05, 32V20
##### Milestones
Received: 16 August 2020
Revised: 6 October 2020
Accepted: 9 October 2020
Published: 14 January 2021
##### Authors
 Ali Maalaoui Department of Mathematics and Computer Science Clark University Worcester, MA United States Department of Mathematics and Natural Sciences American University of Ras Al Khaimah Ras Al Khaimah United Arab Emirates