Deformations of Q-curvature I
arXiv:1512.05389
Abstract
In this article, we investigate deformation problems of -curvature on closed Riemannian manifolds. One of the most crucial notions we use is the -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 -curvature. It includes classifications for nonnegative Einstein -singular spaces, linearized stability of non--singular spaces and a local rigidity result for flat manifolds with nonnegative -curvature. As for global results, we showed that any smooth function can be realized as a -curvature on generic -flat manifolds, while on the contrary a locally conformally flat metric on -tori with nonnegative -curvature has to be flat. In particular, there is no metric with nonnegative -curvature on -tori unless it is flat.
30 pages