A Proof of Uniqueness of the Taub-bolt Instanton

TL;DR

This paper proves that the only Ricci-flat, simply connected 4-manifolds with a specific symmetry and asymptotic flatness are the Schwarzschild and Taub-bolt instantons, establishing their uniqueness under these conditions.

Contribution

It provides a rigorous proof of the uniqueness of the Taub-bolt instanton among Ricci-flat, simply connected 4-manifolds with a symmetry and asymptotic flatness.

Findings

Schwarzschild and Taub-bolt are the only solutions under given conditions.

The proof relies on symmetry and asymptotic flatness assumptions.

The result confirms the special status of these instantons in geometric analysis.

Abstract

We show that the Riemannian Schwarzschild and the ``Taub-bolt'' instanton solutions are the only spaces (M,g) such that 1) M is a 4-dimensional, simply connected manifold with a Riemannian, Ricci-flat C^2-metric g which admits (at least) a 1-parameter group of isometries H without isolated fixed points on M. 2) The quotient (M L)/H (where L is the set of fixed points of H) is an asymptotically flat manifold, and the length of the Killing field corresponding to H tends to a constant at infinity.