Fractional Yamabe solitons and fractional Nirenberg problem

Abstract

In this paper, we first study the fractional Yamabe solitons, which are the self-similar solutions to fractional Yamabe flow. We prove some rigidity results and Liouville type results for such solitons. We then consider the fractional Nirenberg problem: the problem of prescribing fractional order curvature on the sphere. More precisely, we prove that there exists a conformal metric on the unit sphere such that its fractional order curvature is f , when f possesses certain reflection or rotation symmetry.