Non-uniqueness results for critical metrics of regularized determinants in four dimensions

TL;DR

This paper investigates the non-uniqueness and unboundedness of critical metrics related to regularized determinants in four-dimensional conformal geometry, revealing multiple solutions and unbounded actions.

Contribution

It demonstrates non-uniqueness of solutions for conformal metrics and actions, and constructs explicit families of solutions including Delaunay-type solutions.

Findings

Actions are unbounded above and below on conformal four-manifolds.

Existence of a second non-trivial solution on the round sphere.

Construction of periodic solutions, including Delaunay-type solutions.

Abstract

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.