The regularized determinant of the Paneitz operator arises in quantum grav2 ity [see Connes in (Noncommutative geometry, 1994), IV.4.γ ]. An explicit formula for 3 the relative determinant of two conformally related metrics was computed by Branson 4 in (Commun Math Phys 178:301–309, 1996). A similar formula holds for Cheeger’s 5 half-torsion, which plays a role in self-dual field theory [see Juhl in (Families of confor6 mally covariant differential operators, q-curvature and holography. Progress in Mathe7 matics, vol 275, 2009)], and is defined in terms of regularized determinants of the Hodge 8 laplacian on p-forms (p < n/2). In this article we show that the corresponding actions 9 are unbounded (above and below) on any conformal four-manifold. We also show that 10 the conformal class of the round sphere admits a second solution which is not given by 11 the pull-back of the round metric by a conformal map, thus violating uniqueness up to 12 gauge equivalence. These results differ from the properties of the determinant of the 13 conformal Laplacian established in (Commun Math Phys 149:241–262, 1992), (Ann 14 Math 142:171–212, 1995), (Commun Math Phys 189:655–665, 1997). 15 We also study entire solutions of the Euler-Lagrange equation of log det P and the half-torsion τh on R416 \{0}, and show the existence of two families of periodic solutions. 17 One of these families includes Delaunay-type solutions.
Non-uniqueness Results for Critical Metrics of Regularized Determinants in Four Dimensions
MALCHIODI, ANDREA
2012
Abstract
The regularized determinant of the Paneitz operator arises in quantum grav2 ity [see Connes in (Noncommutative geometry, 1994), IV.4.γ ]. An explicit formula for 3 the relative determinant of two conformally related metrics was computed by Branson 4 in (Commun Math Phys 178:301–309, 1996). A similar formula holds for Cheeger’s 5 half-torsion, which plays a role in self-dual field theory [see Juhl in (Families of confor6 mally covariant differential operators, q-curvature and holography. Progress in Mathe7 matics, vol 275, 2009)], and is defined in terms of regularized determinants of the Hodge 8 laplacian on p-forms (p < n/2). In this article we show that the corresponding actions 9 are unbounded (above and below) on any conformal four-manifold. We also show that 10 the conformal class of the round sphere admits a second solution which is not given by 11 the pull-back of the round metric by a conformal map, thus violating uniqueness up to 12 gauge equivalence. These results differ from the properties of the determinant of the 13 conformal Laplacian established in (Commun Math Phys 149:241–262, 1992), (Ann 14 Math 142:171–212, 1995), (Commun Math Phys 189:655–665, 1997). 15 We also study entire solutions of the Euler-Lagrange equation of log det P and the half-torsion τh on R416 \{0}, and show the existence of two families of periodic solutions. 17 One of these families includes Delaunay-type solutions.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
