数学研究所
中国科学院华罗庚数学重点实验室
学术报告
多复变与复几何研讨班
SCV&CG Seminar
Speaker: 王国芳 教授（德国弗莱堡大学）
Title: k-Yamabe problem, revisited
Time&Venue: 2025年3月11日（星期二） 19:00-20:00 & 南楼N913
Abstract: The Yamabe problem, which asks if there exists a conformal metric of constant scalar curvature, plays an important role in the geometric analysis. In the talk we first briefly review this problem. Then we introduce a so-called k-Yamabe problem for a generalized scalar curvature. Its corresponding equation is fully nonlinear and therefore is studied in the so-called k-th Garding cone. This problem has been intensively studied 20 years ago. Now we revisit this problem and study it in a larger cone, the (k-1)-th cone. The results have interesting applications in establishing various Sobolev inequalities. This is a joint work with Yuxin Ge (Toulouse) and Wei Wei (Nanjing and Freiburg).
Title: New optimal Sobolev inequalities
Time&Venue: 2025年3月11日（星期二） 20:10-21:10 & 南楼N913
Abstract: The optimal Sobolev inequality is very important in analysis, especially in the resolution of the Yamabe problem. It has many generalizations, especially to higher order and to fractional order Sobolev inequalities. The optimal Sobolev inequality can also be viewed as an isoperimetric inequality between the total scalar curvature and the volume of a conformal metric. In the talk we will propose a new type Sobolev inequality, which is an isoperimetric inequality relating total higher order Q curvatures. We prove one special case, which already leads to a new Yamabe type problem. At the end I will address this new Yamabe problem. This is a joint work with Yuxin Ge (Toulouse) and Wei Wei (Nanjing and Freiburg).
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