Static vacuum 3+1 black holes that cannot be put into stationary rotation

We prove that some of the static Myers/Korotkin-Nicolai (MKN) vacuum 3+1 static black holes cannot be put into stationary rotation. Namely, they cannot be deformed into axisymmetric stationary vacuum black holes with non-zero angular momentum. We also prove that this occurs in particular for those MKN solutions for which the distance along the axis between the two poles of the horizon is sufficiently small compared to the square root of its area. The MKN solutions, sometimes called periodic Schwarzschild, are physically regular, have no struts or singularities, but are asymptotically Kasner. The static rigidity presented here appears to be the first in the literature of General Relativity.