Dawei Shen

The nonlinear stability of Minkowski spacetime has been one of the central achievements in the mathematical theory of general relativity and, more broadly, in the analysis of nonlinear geometric wave equations. Since the seminal work of Christodoulou-Klainerman, the problem has shaped fundamental advances in our understanding of decay, dispersion, and the intricate interplay between geometry and analysis in the Einstein vacuum equations. This survey presents an overview of the main ideas and techniques underlying the stability theory of Minkowski spacetime. We emphasize the role of decay assumptions, geometric foliations, energy identities, and gauge choices in the global analysis. Particular attention is devoted to exterior stability results, minimal decay regimes, and the borderline case, where the failure of spacetime integrability for nonlinear interactions reveals subtle threshold phenomena. Our goal is to provide a coherent perspective on the evolution of the field, to clarify the structural mechanisms behind the known results, and to outline some of the central open problems that remain in the borderline regime.