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Non-uniqueness results for critical metrics of regularized determinants in four dimensions
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Gursky, Matthew and Malchiodi, A. (2012) Non-uniqueness results for critical metrics of regularized determinants in four dimensions. Communications in Mathematical Physics, Vol.315 (No.1). pp. 1-37. doi:10.1007/s00220-012-1535-7 ISSN 0010-3616.
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Official URL: http://dx.doi.org/10.1007/s00220-012-1535-7
Abstract
The regularized determinant of the Paneitz operator arises in quantum gravity [see Connes in (Noncommutative geometry, 1994), IV.4.γ]. An explicit formula for the relative determinant of two conformally related metrics was computed by Branson in (Commun Math Phys 178:301–309, 1996). A similar formula holds for Cheeger’s half-torsion, which plays a role in self-dual field theory [see Juhl in (Families of conformally covariant differential operators, q-curvature and holography. Progress in Mathematics, vol 275, 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.
Item Type: | Journal Article | ||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
Journal or Publication Title: | Communications in Mathematical Physics | ||||
Publisher: | Springer | ||||
ISSN: | 0010-3616 | ||||
Official Date: | 2012 | ||||
Dates: |
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Volume: | Vol.315 | ||||
Number: | No.1 | ||||
Page Range: | pp. 1-37 | ||||
DOI: | 10.1007/s00220-012-1535-7 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Restricted or Subscription Access |
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