# Yanyan Li

## Rutgers University

### 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.