Abstract

We prove the existence of constant positive σk- scalar curvature metrics which are complete and conformal to the standard metric on Sn\Λ, where Λ⊂Sn is a finite number of points with cardinality at least two, and n, k are positive integers such that 2 ≤ 2k < n. In general this problem is equivalent to solve a singular fully nonlinear second order elliptic equation. For k = 1 (i.e., in the case of the ordinary scalar curvature) the problem reduces to solve a semilinear elliptic equation and it has been studied by several authors (Schoen, Mazzeo-Pacard et al.).