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 graduated from the Institute of Mathematics of the Chinese Academy of Sciences in 1990 and taught at the Institute of Systems of the Chinese Academy of Sciences after graduation. From 1994 to 2006, he was engaged in research at Bochum University in Germany and Max Planck Institute of Applied Mathematics in Germany. From 2006 to 2009, he was a professor at Magdeburg University in Germany, and since 2009, he has been a professor at Freiburg University in Germany. His research direction is geometric analysis and partial differential equations, and he has made a series of important achievements in harmonic mapping, minimal surfaces, Liouville equation, Toda system, k-Yamabe equation, Sasaki Einstein metric, high-order positive mass theorem, geometric inequalities, geometric free boundary value problems, etc., including Duke, CPAM, JDG, JEMS, Crelle’s Journal, Amer. J. Math., Adv. Math., Comm. Math. Phys., Math. Ann.  More than 80 papers have been published in top tier mathematical journals. According to MathScinet statistics, the paper has been cited over 2000 times.