Existence and uniqueness of asymptotically flat toric gravitational instantons

We prove uniqueness and existence theorems for four-dimensional asymptotically flat, Ricci-flat, gravitational instantons with a torus symmetry. In particular, we prove that such instantons are uniquely characterised by their rod structure, which is data that encode the fixed point sets of the torus action. Furthermore, we establish that for every admissible rod structure there exists an instanton that is smooth up to possible conical singularities at the axes of symmetry. The proofs involve adapting the methods that are used to establish black hole uniqueness theorems, to a harmonic map formulation of Ricci-flat metrics with torus symmetry, where the target space is directly related to the metric (rather than auxiliary potentials). We also give an elementary proof of the nonexistence of asymptotically flat toric half-flat instantons. Finally, we derive a general set of identities that relate asymptotic invariants such as the mass to the rod structure.