This is a non-technical introduction to some ideas to derive results in scalar curvature geometry and general relativity using also singular solutions of variational problems. A typical class of such solutions are area minimizing hypersurfaces. They are known to admit complicated singular sets. We will see how, even without knowing the structure of these singular loci, Gromov hyperbolic geometry gives us a fine control over the asymptotic analysis of elliptic operators on such hypersurfaces towards their singularities.