Imagine stretching an n-dimensional ellipsoid inside Euclidean space so that one of its semiaxes becomes very large. This causes a corresponding stretching of geometric objects that locally minimize length (geodesics) or area (minimal surfaces) inside that ellipsoid, making them less stable. In this talk, I will explain how this growing instability can be exploited to detect hard-to-find solutions to these variational problems that evade most cutting-edge methods of Geometric Analysis.