The functional determinant of the Paneitz operator

Details The functional determinant of the Paneitz operator The functional determinant of the Paneitz operator

2011-05-18

arxiv.org/abs/1105.3762v1

Mathematics

Analysis of PDEs

35 pages

Scientific paper

The regularized determinant of the Paneitz operator arises in quantum gravity (see [Con94], IV.4.). An explicit formula for the relative determinant of two conformally related metrics was computed by Branson in [Bra96]. In this article we show that the corresponding action is unbounded (above and below) on any conformal four-manifold. We also show that the conformal class of the round sphere admits a second solution which is not given by the pull-back of the round metric by a conformal map, thus violating uniqueness up to gauge equivalence. These results differ from the properties of the determinant of the conformal Laplacian established in [CY95], [BCY92], and [Gur97]. We also study entire solutions of the Euler-Lagrange equation of log det P on R4 \ {0}, and show the existence of two families of periodic solutions. One of these families includes Delaunay-type solutions.

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