Branson's Q-curvature in Riemannian and Spin Geometry

Oussama Hijazi (1) , Simon Raulot (2)

1 IECN - Institut Élie Cartan de Nancy
2 UNINE - Institut de Mathématiques

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. On a closed n-dimensional manifold, $n\ge 5$, we compare the three basic conformally covariant operators : the Branson-Paneitz, the Yamabe and the Dirac operator (if the manifold is spin) through their first eigenvalues. Equality cases are also characterized.

Mots clés: Branson's Q-curvature; Eigenvalues; Yamabe operator; Branson-Paneitz operator; Dirac operator; $\sigma_k$-curvatures; Yamabe invariant; Conformal Geometry; Einstein metrics; Killing spinors

Domaines: Géométrie différentielle [math.DG]

Oussama Hijazi, Simon Raulot. Branson's Q-curvature in Riemannian and Spin Geometry. 2007. ⟨hal-00169447v2⟩