01662nas a2200121 4500008004100000245009600041210006900137260001300206520124300219100002001462700002201482856003601504 2012 en d00aNon-uniqueness results for critical metrics of regularized determinants in four dimensions0 aNonuniqueness results for critical metrics of regularized determ bSpringer3 aThe 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.1 aGursky, Matthew1 aMalchiodi, Andrea uhttp://hdl.handle.net/1963/655900729nas a2200121 4500008004300000245009900043210006900142260001300211520030900224100002200533700001600555856003600571 2011 en_Ud 00aNew improved Moser-Trudinger inequalities and singular Liouville equations on compact surfaces0 aNew improved MoserTrudinger inequalities and singular Liouville bSpringer3 aWe consider a singular Liouville equation on a compact surface, arising from the study of Chern-Simons vortices in a self dual regime. Using new improved versions of the Moser-Trudinger inequalities (whose main feature is to be scaling invariant) and a variational scheme, we prove new existence results.1 aMalchiodi, Andrea1 aRuiz, David uhttp://hdl.handle.net/1963/409900389nas a2200109 4500008004100000245007400041210006900115260001300184100002400197700002200221856003600243 2001 en d00aNon-compactness and multiplicity results for the Yamabe problem on Sn0 aNoncompactness and multiplicity results for the Yamabe problem o bElsevier1 aBerti, Massimiliano1 aMalchiodi, Andrea uhttp://hdl.handle.net/1963/134500420nas a2200121 4500008004100000245007300041210006900114260001800183100002400201700001500225700002200240856003600262 2000 en d00aA note on the scalar curvature problem in the presence of symmetries0 anote on the scalar curvature problem in the presence of symmetri bSISSA Library1 aAmbrosetti, Antonio1 aYanYan, Li1 aMalchiodi, Andrea uhttp://hdl.handle.net/1963/1365