Some recent works on conformally invariant fully nonlinear elliptic equations
The following problem was raised by Nirenberg: Which function on the standard 2-sphere is the Gauss curvature of a metric conformally equivalent to the standard metric. Naturally one may ask a similar question in the higher dimensional case, namely, which function on the standard \(n\)-sphere is the scalar curvature of a metric conformally equivalent to the standard metric.
An analogous question can be asked for the \(\sigma_k\) curvature instead of the scalar curvature, and we call it the \(\sigma_k\)-Nirenberg problem. We will present some results on the existence and compactness of solutions of the \(\sigma_k\)-Nirenberg problem for \(n≥ 3\) and \(k≥ n/2\). The results for \(n=4\) and \(k=2\) were established by Alice Chang, Zheng-Chao Han and Paul Yang in 2011. We will also present some recent results on the \(σ_k\)-Loewner–Nirenberg problem. These results are from a couple of joint works with Maria del Mar Gonzalez, Luc Nguyen, Bo Wang, and Jingang Xiong.