Deformations of Q-curvature I

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.