Branson's Q-curvature in Riemannian and Spin Geometry

TL;DR

This paper explores relationships between Branson's Q-curvature, eigenvalues of conformally covariant operators, and geometric properties of closed Riemannian and spin manifolds, providing bounds and characterizations of equality cases.

Contribution

It establishes bounds for eigenvalues of key operators in terms of Q-curvature and characterizes when equality occurs, extending understanding in conformal and spin geometry.

Findings

Lower bound for Yamabe operator eigenvalue in terms of Q-curvature

Relation between Dirac operator eigenvalue and Q-curvature on spin manifolds

Comparison of eigenvalues of conformally covariant operators in higher dimensions

Abstract

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.