Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 3 (2007), 119, 11 pages      arXiv:0709.0345      https://doi.org/10.3842/SIGMA.2007.119
Contribution to the Proceedings of the 2007 Midwest Geometry Conference in honor of Thomas P. Branson

Branson's Q-curvature in Riemannian and Spin Geometry

Oussama Hijazi and Simon Raulot
Institut Élie Cartan Nancy, Nancy-Université, CNRS, INRIA, Boulevard des Aiguillettes B.P. 239 F-54506 Vandoeuvre-lès-Nancy Cedex, France

Received August 25, 2007, in final form November 29, 2007; Published online December 11, 2007
Section 5 regarding the relation with the σ2-curvature has been removed and Remark 1 (page 9) has been added January 27, 2008.

Abstract
On a closed n-dimensional manifold, n ≥ 5, we compare the three basic conformally covariant operators: the Paneitz-Branson, the Yamabe and the Dirac operator (if the manifold is spin) through their first eigenvalues. On a closed 4-dimensional Riemannian manifold, we give a lower bound for the square of the first eigenvalue of the Yamabe operator in terms of the total Branson's Q-curvature. As a consequence, if the manifold is spin, we relate the first eigenvalue of the Dirac operator to the total Branson's Q-curvature. Equality cases are also characterized.

Key words: Branson's Q-curvature; eigenvalues; Yamabe operator; Paneitz-Branson operator; Dirac operator; σk-curvatures; Yamabe invariant; conformal geometry; Killing spinors.

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