ISOLATION OF GRAVITATIONAL INSTANTONS - FLAT TORI VERSUS FLAT R4

The role of topology in the perturbative solution of the Euclidean Ein stein equations (EEE's) about flat instantons is examined. When the to pology is open (with asymptotically flat boundary conditions) it is si mple to demonstrate that all vacuum perturbations vanish at all orders in perturbation theory; when the topology is closed (a four-torus say ) all but a 10-parameter family of global metric deformations (moving us from one flat torus to another) vanish. Flat solutions, regardless of their topology, are perturbatively isolated as solutions of the EEE 's. The perturbation theory of the complete Einstein equations contras ts dramatically with that of the trace of these equations, the vanishi ng of the scalar curvature. In the latter case, the flat tori are isol ated whereas R4 is not. This is a consequence of a linearization insta bility of the trace equation which is not a linearization instability of the complete EEE's.