Mathematics

The regularized determinant of the Paneitz operator arises in quantum gravity
(see Connes 1994, IV.4.$\gamma$). An explicit formula for the relative
determinant of two conformally related metrics was computed by Branson in
Branson (1996). A similar formula holds for Cheeger's half-torsion, which plays
a role in self-dual field theory (see Juhl, 2009), and is defined in terms of
regularized determinants of the Hodge laplacian on $p$-forms ($p < n/2$). In
this article we show that the corresponding actions are 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 Chang and Yang (1995), Branson, Chang, and
Yang (1992), and Gursky (1997).
We also study entire solutions of the Euler-Lagrange equation of $\log \det
P$ and the half-torsion $\tau_h$ on $\mathbb{R}^4 \setminus {0}$, and show the
existence of two families of periodic solutions. One of these families includes
Delaunay-type solutions.