Harmonic maps in singular geometry and rigidity

This survey reviews results on harmonic maps into spaces of non-positive curvature, with a focus on targets that lack smooth structure. More precisely, we consider targets that are complete metric spaces with non-positive curvature in the sense of Alexandrov, commonly referred to as NPC (non-positively curved) or CAT(0) spaces. We discuss applications of harmonic maps to rigidity phenomena, including generalizations of Margulis superrigidity and the holomorphic rigidity of Teichm\"uller space. Our approach relies heavily on the regularity theory of harmonic maps to non-smooth targets, enabling differential-geometric techniques to be employed in the absence of any smooth structure on the target.