偏微分方程研讨班：A YAMABE PROBLEM FOR THE QUOTIENT BETWEEN THE Q CURVATURE AND THE SCALAR CURVATURE  

中国科学院数学与系统科学研究院

数学研究所

数学科学全国重点实验室

偏微分方程研讨班

Speaker: 韦韡 助理教授（南京大学）

Inviter: 邱国寰

Language: Chinese

Title: A YAMABE PROBLEM FOR THE QUOTIENT BETWEEN THE Q CURVATURE AND THE SCALAR CURVATURE  

Time & Venue: 2026年6月17日（星期三）17：00-18：00 &思源楼S813

Abstract:In this talk we introduce the following Yamabe problem for the quotient between the $Q$ curvature and the scalar curvature $R$: Find a conformal metric $g$ in a given conformal class $[g_0]$ with Q_g\slash R_g=const.We first prove a new Sobolev inequality between the total $Q$-curvature and the total scalar curvature for any $g$ in the conformal class of the round metric $\gSn$ with positive scalar curvature, with equality if and only if $g$ is also a metric with constant sectional curvature. With this inequality we introduce a new Yamabe constant $Y_{4,2}(M,[g_0])$ and prove the existence of the above problem provided that $Y_{4,2}(M,[g_0]) <Y_{4,2} (\ss^n, [g_{\ss^n}]).$ This strict inequality is proved if $(M,g)$ is not conformally equivalent to the round  sphere. This follows from  a crucial relation between $Y_{4,2}$ and the ordinary Yamabe constant $Y(M,[g_0])$, which is proved in this paper, and the resolution of the ordinary Yamabe problem.Finally, we prove that on a closed $n$-dimensional  Riemannian manifold $(M,g_{0})$ with semi-positive $Q$-curvature and non-negative scalar curvature, the above Yamabe problem is solvable, thanks to the maximum principle of Gursky-Malchiodi, when $n\ge 5$. The proof for $n=3$ and $n=4$ follows closely the methods developed by Hang-Yang Gursky-Malchiodi, and Chang-Yang. This is a joint work with Y.X. Ge and G.F. Wang。