Walter Simon: Gravitational Instantons with S^1 Symmetry

Date: 2023-07-10

Time: 11:20 - 12:00

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Walter Simon, University of Vienna


We take Gravitational Instantons M to be smooth Ricci-flat four-manifolds with at least quadratic curvature decay. Attemps for their classification involve more specific requirements on the asymptotic falloff of the metric, and/or assumptions on topology, symmetry, Hermiticity, the (hy-per) kähler property, and algebraic properties of the Weyl tensor. Known examples include Kerr, Taub-NUT, Taub-Bolt and the Chen-Teo instantons.

We contribute to this classification by proving the following result for asymptotically (locally or globally) flat S1-instantons:
1. If M ∼= S^4 S^1, then M is Kerr.
2. If M ∼= S^4 {∞}, then M is Taub-Bolt.
3. If M ∼= CP^2 S1, then M is Hermitian.

Key ingredients of the proof are the index theorem and the G-signature theorem applied to the S1 Killing field, as well as a pair of Rellich-Pohozaev-Israel- type of divergence identities.

This is joint work with Steffen Aksteiner, Lars Andersson, Mattias Dahl, and Gustav Nilsson.