In this talk, we will first review the resolution of the Chang-Yang conjecture on a sharp Trudinger-Moser inequality on the two-dimensional sphere. Then higher dimensional analogues will be presented and some very recent progress will be introduced. We will also mention some related open problems that the audience might be interested in.
The talk is based on joint work with Changfeng Gui (University of Macau) and Weihong Xie (Central South University).