Rigidity of compact static near-horizon geometries with negative cosmological constant

In this note, we show that compact static near-horizon geometries with negative cosmological constant are either Einstein or the product of a circle and an Einstein metric. Chruściel, Reall, and Todd proved rigidity when the cosmological constant vanishes, in which case one get the stronger result that the space is Ricci flat (Chruściel et al. in Class Quantum Gravity 23:549–554, 2006). It has been previously asserted that a stronger rigidity statement also holds for negative cosmological constant, but Bahuaud, Gunasekaran, Kunduri, and Woolgar recently pointed out that this was not the case (Bahuaud et al. in Lett Math Phys 112(6):116, 2022). They showed, moreover, that for a compact static near-horizon geometry with negative cosmological constant, the potential vector field X is constant length and divergence-free. We give an argument using the Bochner formula to improve their conclusion to X being a parallel field, which implies the optimal rigidity result. The result also holds more generally for m-Quasi Einstein metrics with m> 0.