Functional determinant on pseudo-Einstein 3-manifolds

Ali Maalaoui

Vol. 309 (2020), No. 2, 421–436

DOI: 10.2140/pjm.2020.309.421

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 $𝜃\mapsto e^w𝜃$ with $w\in \mathcal{𝒫}$ 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