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Gravitational instantons with $S^1$ symmetry

Steffen Aksteiner, Lars Andersson, Mattias Dahl, Gustav Nilsson, Walter Simon

TL;DR

The paper addresses the classification of ALF $S^1$-symmetric gravitational instantons by combining a divergence identity with the $G$-signature theorem. The authors prove an $S^1$-symmetric Euclidean Black Hole Uniqueness result, establish a Taub-bolt uniqueness, and show Hermitianity for ALF instantons with topology $\,\mathbb{C}P^2\setminus S^1$, guided by fixed-point data (nuts and bolts) and asymptotic invariants. The approach hinges on translating geometric constraints into boundary contributions and exploiting algebraic speciality via Ernst potentials and Simon-type currents, yielding rigidity results that align with known Kerr and Taub-bolt families and the Chen--Teo scenario. This provides a robust framework for rigidity and uniqueness in gravitational instanton geometry and points toward broader applicability of the $G$-signature-based strategy for $S^1$-actions. Overall, the work advances the classification program for ALF gravitational instantons and clarifies the role of symmetry, topology, and Hermitian structure in geometric rigidity.

Abstract

Uniqueness results for asymptotically locally flat and asymptotically flat $S^1$-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 $S^1$-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 $S^1$-symmetric instanton with the topology of the Chen-Teo family of instantons is Hermitian.

Gravitational instantons with $S^1$ symmetry

TL;DR

The paper addresses the classification of ALF -symmetric gravitational instantons by combining a divergence identity with the -signature theorem. The authors prove an -symmetric Euclidean Black Hole Uniqueness result, establish a Taub-bolt uniqueness, and show Hermitianity for ALF instantons with topology , guided by fixed-point data (nuts and bolts) and asymptotic invariants. The approach hinges on translating geometric constraints into boundary contributions and exploiting algebraic speciality via Ernst potentials and Simon-type currents, yielding rigidity results that align with known Kerr and Taub-bolt families and the Chen--Teo scenario. This provides a robust framework for rigidity and uniqueness in gravitational instanton geometry and points toward broader applicability of the -signature-based strategy for -actions. Overall, the work advances the classification program for ALF gravitational instantons and clarifies the role of symmetry, topology, and Hermitian structure in geometric rigidity.

Abstract

Uniqueness results for asymptotically locally flat and asymptotically flat -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 -signature theorem. Our results include a proof of the -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 -symmetric instanton with the topology of the Chen-Teo family of instantons is Hermitian.
Paper Structure (25 sections, 36 theorems, 240 equations)

This paper contains 25 sections, 36 theorems, 240 equations.

Table of Contents

  1. Introduction
  2. ALF instantons
  3. The fixed point set
  4. Topological invariants
  5. Structure of the fixed point set
  6. Geometry with a Killing field
  7. Charges
  8. Divergence identities
  9. Characterizations of special geometry
  10. Integrating the divergence identity
  11. Contribution from the fixed point set
  12. Contribution from infinity
  13. The singular term
  14. Proof of Proposition \ref{['prop:magbound']}
  15. Proof of the main theorem
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $(\mathcal{M}, g_{ab})$ be an $S^1$-instanton of ALF-$A_k$ type. Then the following holds.

Theorems & Definitions (99)

  • Theorem 1.1
  • Conjecture 1: MR4809333
  • Remark 1.2
  • Conjecture 2
  • Remark 1.3
  • Conjecture 3
  • Remark 1.4
  • Definition 2.1: Biquard:Gauduchon
  • Definition 2.2: ALF instanton
  • Remark 2.3
  • ...and 89 more