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 …