ar X iv : g r - qc / 9 80 90 76 v 1 2 8 Se p 19 98 Bolts have no hair

We show that the " Taub-bolt instantons " are the only Riemannian, Ricci-flat and asymptotically locally flat (ALF) C 2-metrics on a 4-dimensional, complete, simply connected man-ifold having (at least) a one-parameter group of isometries which are compatible with ALF, have trajectories of bounded length at infinity and no isolated fixed points. Attempts of estimating the path integral of Quantum Gravity via the stationary phase approximation motivated the study of " instantons " , i.e. Riemannian, Ricci-flat metrics which are regular everywhere. Gibbons and Hawking [1] classified instantons having at least a one-parameter group of isometries according to whether the isometry has isolated fixed points (" nuts ") or two-dimensional subsets of fixed points (" bolts "). The mathematical analogy between instantons with fixed points and Lorentzian solutions with Killing horizons raises the question whether the known uniqueness proofs for black hole solutions can be carried over to the Riemannian case. In fact, this works straightforwardly (and yields the Riemannian Schwarzschild solution) when the solution is assumed to be asymptotically flat (AF) and when the Killing field is hypersurface orthogonal. For AF solutions with up to two nuts and no bolts, a uniqueness result for the Riemannian Kerr metric has been obtained [2] by adapting and generalizing Israel's proof of uniqueness of the Schwarzschild solution [3]. In this paper we present the following uniqueness result for the so-called Taub-bolt instanton (found by Page [4] by " Euclideanizing " the Taub-NUT spacetime [5]).