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