Isolation of gravitational instantons: Flat tori versus flat R4

The role of topology in the perturbative solution of the Euclidean Einstein equations (EEE's) about flat instantons is examined. When the topology is open (with asymptotically flat boundary conditions) it is simple 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 contrasts dramatically with that of the trace of these equations, the vanishing of the scalar curvature. In the latter case, the flat tori are isolated whereas R⁴ is not. This is a consequence of a linearization instability of the trace equation which is not a linearization instability of the complete EEE's.