Nonexistence of degenerate horizons in static vacua and black hole uniqueness

We show that in any spacetime dimension D≥4, degenerate components of the event horizon do not exist in static vacuum configurations with positive cosmological constant. We also show that without a cosmological constant asymptotically flat solutions cannot possess a degenerate horizon component. Several independent proofs are presented. One proof follows easily from differential geometry in the near-horizon limit, while others use Bakry-Émery-Ricci bounds for static Einstein manifolds.