Nuts have no hair

We show that the Riemannian Kerr solutions are the only Riemannian, Ricci-flat and asymptotically flat ${\mathrm{C}}^2$-metrics $g_{\mu \nu }$ on a 4-dimensional complete manifold $\mathcal{M}$ of topology ${\rm R}^{2} \times {\rm S}^{2}$ which have (at least) a 1-parameter group of periodic isometries with only isolated fixed points (“nuts”) and with orbits of bounded length at infinity.