## Branson's $Q$-curvature in Riemannian and spin geometry

### Summary

Summary: On a closed $n$-dimensional manifold, $n \geq 5$, 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.

### Mathematics Subject Classification

53C20, 53C27, 58J50

### Keywords/Phrases

branson's $Q$-curvature, eigenvalues, yamabe operator, Paneitz-Branson operator, Dirac operator, $\sigma _{k}$-curvatures, yamabe invariant, conformal geometry, Killing spinors