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ShowMe Analysis Meeting 2005 |
Thomas Branson, University of Iowa
Q-curvature and spectral invariants
Abstract: The Q-curvature on an even-dimensional Riemannian manifold is a local invariant whose properties generalize those of the 2-dimensional Gauss curvature. In particular, it provides geometric analysts with a natural exponential class conformal prescription problem. In addition, the Q-curvature occurs in Nature,
- in functional determinant quotient formulas for pairs of metrics in a conformal class, and in the related exponential class inequality that estimates these determinants;
- (as shown in recent work of Graham and Hirachi) as the action principle whose total metric variation is the Fefferman-Graham obstruction tensor, thus providing a generalization of Weyl relativity from four to arbitrary even dimensions.
Recent joint work with Rod Gover shows that there is additional, heretofore hidden structure in the de Rham complex, from the viewpoint of conformal structure: higher order analogues of the Maxwell operator and its Eastwood-Singer gauge operator, in which the information at different orders interacts. Part of this picture is a sequence of objects that act like the Q-curvature. On the spectral side, we get Polyakov-type formulas for "detour torsions,'' the simplest of which coincides with the Cheeger half-torsion.