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 -
TY - JOUR
T1 - New improved Moser-Trudinger inequalities and singular Liouville equations on compact surfaces
JF - Geometric and Functional Analysis 21 (2011) 1196-1217
Y1 - 2011
A1 - Andrea Malchiodi
A1 - David Ruiz
AB - We 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.
PB - Springer
UR - http://hdl.handle.net/1963/4099
U1 - 305
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - JOUR
T1 - Non-compactness and multiplicity results for the Yamabe problem on Sn
JF - J. Funct. Anal. 180 (2001) 210-241
Y1 - 2001
A1 - Massimiliano Berti
A1 - Andrea Malchiodi
PB - Elsevier
UR - http://hdl.handle.net/1963/1345
U1 - 3110
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - JOUR
T1 - A note on the scalar curvature problem in the presence of symmetries
JF - Ricerche Mat. 49 (2000), suppl., 169-176
Y1 - 2000
A1 - Antonio Ambrosetti
A1 - Li YanYan
A1 - Andrea Malchiodi
PB - SISSA Library
UR - http://hdl.handle.net/1963/1365
U1 - 3090
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -