Deformations of Q-curvature I

Abstract

In this article, we investigate deformation problems of $Q$-curvature on closed Riemannian manifolds. One of the most crucial notions we use is the $Q$-singular space, which was introduced by Chang-Gursky-Yang during 1990's. Inspired by the early work of Fischer-Marsden, we derived several results about geometry related to $Q$-curvature. It includes classifications for nonnegative Einstein $Q$-singular spaces, linearized stability of non-$Q$-singular spaces and a local rigidity result for flat manifolds with nonnegative $Q$-curvature. As for global results, we showed that any smooth function can be realized as a $Q$-curvature on generic $Q$-flat manifolds, while on the contrary a locally conformally flat metric on $n$-tori with nonnegative $Q$-curvature has to be flat. In particular, there is no metric with nonnegative $Q$-curvature on $4$-tori unless it is flat.