Hawking's local rigidity theorem without analyticity

Understand

We 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.