Deformations of Q-curvature I

TL;DR

This paper studies the deformation of Q-curvature on closed Riemannian manifolds, providing classifications, stability results, and global realizability conditions, advancing understanding of geometric structures related to Q-curvature.

Contribution

It introduces new classification and stability results for Q-singular spaces and establishes global and local rigidity theorems for manifolds with nonnegative Q-curvature.

Findings

Classified nonnegative Einstein Q-singular spaces

Proved local rigidity for flat manifolds with nonnegative Q-curvature

Showed that any smooth function can be realized as Q-curvature on generic Q-flat manifolds

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.