In recent years there has been considerable interest in the so-called singular Yamabe problem: given a compact Riemannian manifold \((M,g_0)\) of dimension \(N\geq 3\) and a closed set \(\Lambda\subset M\), find a metric \(g\) conformal to \(g_0\) such that \(g\) has constant scalar curvature and \(g\) is complete on \(M\backslash\Lambda\). If the new metric is given by \(g =u^{4/(N-2)}g_0\) for a smooth positive function \(u\) on \(M\backslash\Lambda\), then \(u\) is required to satisfy the equation \(\Delta_{g_0}u - {{N-2}\over{4(N-1)}} R(g_0)u + {{N-2}\over{4(N-1)}}R(g) u^{(N+2)/(N-2)} = 0\) in \(M\backslash\Lambda\), where \(R(g_0)\), \(R(g)\) are the scalar curvatures of \(g_0\), \(g\) respectively, and \(\Delta_{g_0}\) is the Laplace-Beltrami operator of \(g_0\). In addition, for \(g\) to be complete on \(M\backslash\Lambda\), \(u\) must tend to infinity sufficiently fast as \(\Lambda\) is approached.
It is known that for a solution of the above problem to exist the size of \(\Lambda\) and the sign of \(R=R(g)\) must be related. If a solution exists for \(R<0\), then \(\text{ dim}(\Lambda)>(N-2)/2\), while if a solution exists for \(R\geq 0\), then \(\text{ dim}(\Lambda)\leq (N-2)/2\) and the first eigenvalue of \(\Delta_{g_0}-{{N-2}\over{4(N-1)}}R(g_0)\) must be nonnegative. A variety of partial converses have been proved. In particular, in [J. Differ. Geom. 44, 331-370 (1996; Zbl 0869.35040)] the authors proved the existence of a solution for \(M\) an arbitrary compact manifold of nonnegative scalar curvature, whenever \(\Lambda\) is a disjoint finite union of submanifolds of dimensions between \(1\) and \((N-2)/2\). Here this is extended, for \((M,g_0)={\mathbb S}^N\) with the standard metric, to allow \(\Lambda\) to be a disjoint finite union of submanifolds of dimensions between zero and \((N-2)/2\).