Gravitational instantons with 𝑆¹ symmetry

Abstract Uniqueness results for asymptotically locally flat and asymptotically flat <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>S</m:mi> <m:mn>1</m:mn> </m:msup> </m:math> S^{1} -symmetric gravitational instantons are proved using a divergence identity of the type used in uniqueness proofs for static black holes, combined with results derived from the 𝐺-signature theorem. Our results include a proof of the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>S</m:mi> <m:mn>1</m:mn> </m:msup> </m:math> S^{1} -symmetric version of the Euclidean Black Hole Uniqueness Conjecture, a uniqueness result for the Taub-bolt family of instantons, as well as a proof that an ALF <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>S</m:mi> <m:mn>1</m:mn> </m:msup> </m:math> S^{1} -symmetric instanton with the topology of the Chen–Teo family of instantons is Hermitian.