In [Progress Math.233 (2005)], David suggested the existence of a new type of global minimizers for the Mumford-Shah functional in ${𝐑}^3$. The singular set of such a new minimizer belongs to a three parameters family of sets $(0<{\delta }_1,{\delta }_2,{\delta }_3<\pi )$. We first derive necessary conditions satisfied by global minimizers of this family. Then we are led to study the first eigenvectors of the Laplace-Beltrami operator with Neumann boundary conditions on subdomains of ${𝐒}^2$ with three reentrant corners. The necessary conditions are...