Abstract: We consider a new 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 $Q_g / R_g=const$. When the dimension $n\geq5$, it relates to a new Sobolev inequality between the total $Q$-curvature and the total scalar curvature on $\mathbb{S}^n$ ($n\geq5$). 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 $Y_{4,2}(M,[g_0]) <Y_{4,2}(\mathbb{S}^n,[g_{\mathbb{S}^n}])$ by a flow method. This strict inequality is proved if $(M,g)$ is not conformally equivalent to the round sphere. This is a joint work with Wei Wei and Guofang Wang.

Brief biography: Yuxin Ge is currently a professor at University of toulouse. He obtained his PhD at école normale supérieure de Cachan in 1997 under the supervision of Frédéric Hélein. His research focuses on geometric analysis and elliptic PDE.

Zoom: link          ID: 893 4998 4451           Password: 514608