We study construction of a complete non-compact Riemannian metrics with positive constant \(σ_2\)-curvature, on the sphere \(\mathbb S^n\) with a prescribed singular set \(\Lambda\) given by a disjoint union of closed submanifolds whose dimension is positive and strictly less than \(\frac{n−\sqrt{ n}−2}{2}\). This is a fully non-linear problem, nevertheless, we show that the classical gluing method of Mazzeo-Pacard for the scalar curvature still works in this setting since the linearized operator has good mapping properties in weighted spaces. A necessary condition is that \(2 ≤ 2k < n\).

Joint work with M. Del Mar González and Lorenzo Mazzieri.