2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/219055We prove the existence of a Hawking Killing vector-field in a full neighborhood of a local, regular, bifurcate, non-expanding horizon embedded in a smooth vacuum Einstein space-time. We do not assume analyticity of the space-time. This result will be applied in a second paper to prove a perturbative version of the uniqueness of smooth, stationary black holes in vacuum.General Relativity and Quantum CosmologyAnalysis of PDEstext