We first recall the $\sigms_k$ Yamabe problem and study it in a larger cone. The corresponding Sobolev inequality holds then in the larger cone. We also prove a new type Sobolev inequality and consider a new type Yamabe problem involving $Q$-curvatures, which leads to many open problems. The talk bases on joint work with Yuxin Ge and Wei Wei.

Speaker Biography：He earned his Ph.D. from the Institute of Mathematics, Chinese Academy of Sciences in 1990 and subsequently taught at the Academy of Mathematics and Systems Science, Chinese Academy of Sciences. From 1994 to 2006, he conducted research at Ruhr University Bochum and the Max Planck Institute for Mathematics in the Sciences in Germany. He served as a professor at the University of Magdeburg from 2006 to 2009 and has been a professor at the University of Freiburg since 2009.

His research focuses on geometric analysis and partial differential equations, with significant contributions to harmonic maps, minimal surfaces, the Liouville equation, Toda systems, the k-Yamabe equation, Sasaki–Einstein metrics, the higher-order positive mass theorem, geometric inequalities, and geometric free boundary problems. He has published over 80 papers in leading mathematical journals, including Duke Mathematical Journal, Communications on Pure and Applied Mathematics, Journal of Differential Geometry, Journal of the European Mathematical Society, Crelle's Journal, American Journal of Mathematics, Advances in Mathematics, Communications in Mathematical Physics, and Mathematische Annalen. According to MathSciNet, his work has been cited more than 2,000 times.