TY - JOUR
T1 - Non-uniqueness results for critical metrics of regularized determinants in four dimensions
JF - Communications in Mathematical Physics, Volume 315, Issue 1, September 2012, Pages 1-37
Y1 - 2012
A1 - Matthew Gursky
A1 - Andrea Malchiodi
AB - 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.
PB - Springer
UR - http://hdl.handle.net/1963/6559
N1 - 35 pages, title changed, added determinant of half-torsion,
references added. Comm. Math. Phys., to appear
U1 - 6488
U2 - Mathematics
U4 - 1
U5 - MAT/05 ANALISI MATEMATICA
ER -