Singular solutions for the constant \(Q\)-curvature problem (Q2215826)

The \(Q\)-curvature of a smooth Riemannian manifold \((M,g)\) of dimension \(n>2\) is \[ Q=-\Delta J-2|A|^2+\tfrac12nJ^2, \] where \(\Delta\) is the Laplace opeator, \(R\) is the scalar curvature, \(J=\frac1{2n-2}R\) and \(A=\frac1{n-2}(Ric-Jg)\). The main result of the paper is the following Theorem. Let \(M\) be a connected closed manifold of dimension \(n\geq5\) and \(\Sigma\) its connected closed submanifold of dimension \(0< k<\frac{n}{2}-2\). If \(M\) admits \(g\) with \(Q\geq0\), \(Q\not\equiv0\), \(R\geq0\), then there exists an infinite-dimensional family of complete metrics on \(M\setminus\Sigma\) with constant \(Q\)-curvature. The proof is based on the existence result for positive solutions to the PDE \(\Delta^2 u=u^p\) on a domain \(\Omega\subset\mathbb{R}^n\) outside a finite union of submanifolds \(\Sigma_i\), with \(u=\Delta u=0\) on \(\partial\Omega\) and \(u\) blowing up on \(\Sigma=\cup\Sigma_i\) provided the following inequalities hold for \(p\) and dimensions \(k_i\) of the components \(\Sigma_i\) of \(\Sigma\): \[ 4p< (p-1)(n-k_i)< 4(p+1). \] This gives a construction of singular metrics on closed manifolds with constant positive \(Q\)-curvature.