Abstract:
In the classic Yamabe problem, we want to find  a constant scalar curvature metric in each conformal metric class.  $\sigma _k$ Yamabe problem is a fully nonlinear version of  the Yamabe problem. Instead of constant scalar curvature, we would like to find a constant $\sigma_k$ curvature metric in each conformal class. In this talk, I will focus on $\sigma_2$ Yamabe problem in 4 dimension. I’ll brief the motivation behind this problem and the existence result. I then plan to discuss the recent uniqueness result by Gursky-Streets using the formal metric structure of the conformal class inspired by the Mabuchi–Semmes–Donaldson metric for Kaehler class.