Lectures on the fourth-order \(Q\) curvature equation (Q2815945)

The goal of the reviewed paper is to find out the role of the Paneitz operator and the \(Q\) curvature in understanding the geometry of a conformal class and the topology of the underlying manifold.NEWLINENEWLINELet \((M,g)\) be a smooth Riemannian manifold with dimension \(n\geqslant 3\). The fourth-order \(Q\) curvature is given by NEWLINE\[NEWLINEQ=-\Delta J-2|A|^2+\frac n2 J^2,NEWLINE\]NEWLINE where \(J=\frac{R}{2(n-1)},\;A=\frac{1}{n-2}(Rc-Jg)\), \(R\) and \(Rc\) denote the scalar curvature and the Ricci tensor.NEWLINENEWLINEThe Paneitz operator is defined as NEWLINE\[NEWLINEP\phi=\Delta^2\phi+\operatorname{div}(4A(\nabla\phi,e_i)e_i-(n-2)J\nabla\phi)+\frac{n-4}{2}Q\phi\,.NEWLINE\]NEWLINE The authors discuss recent progress in dimension \(n\geqslant 5\) about the Green's function of the Paneitz operator and the solution to finding constant \(Q\) curvature in a fixed conformal class. Next they consider the case \(n=3\), where the \(Q\) curvature equation is particularly intriguing. Moreover, the authors present many open problems.NEWLINENEWLINEFor the entire collection see [Zbl 1341.53003].