Toric gravitational instantons with four fixed points
Abstract
We classify the simply connected toric gravitational instantons whose torus action has four fixed points. They are the hyperkähler multi-Eguchi--Hanson and multi-Taub--NUT metrics with four collinear centres, the asymptotically flat instantons on Li and Sun's manifold X3, up to scale one for each twist at infinity in (0,1) for a fixed orientation of the torus, and, up to scale, a single further instanton with an ALF end, which is not hyperkähler. The last two are Hermitian for neither orientation. In particular, the toric asymptotically flat instanton on X3 with a given twist is unique up to scale, as conjectured by Li and Sun. The proof combines the solitonic form of Geroch's conjecture, which turns smoothness into polynomial equations for the soliton data, with a convexity principle adapted from Aksteiner. At a fixed asymptotic parameter, the cone errors of the conical metrics with a given rod structure are the gradient of a strictly convex function of the rod lengths, and the strict convexity gives uniqueness. Existence follows from an explicit solution for the ALF end and, for the AF ends, from Li and Sun's existence theorem. We also outline how a chart of Aksteiner identifies these AF instantons with those found by Teo.
Related
- The twistor construction of toric gravitational instantons
- On the Geroch conjecture for toric gravitational instantons
- Unitary S1-Manifolds with Four Fixed Points and Kosniowski's Conjecture
- Four point analogue of Ciric-Reich-Rus type mappings with non-unique fixed points
- Gravitational instantons and hyperbolic space
BibTeX
@misc{andersson2026toric,
title = {{Toric gravitational instantons with four fixed points}},
author = {Lars Andersson and Mattias Dahl and Gustav Nilsson},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05333},
eprint = {2611.05333},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05333},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Toric gravitational instantons with four fixed points
AU - Lars Andersson
AU - Mattias Dahl
AU - Gustav Nilsson
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.05333
UR - https://arxiv.org/abs/2611.05333
AB - We classify the simply connected toric gravitational instantons whose torus action has four fixed points. They are the hyperkähler multi-Eguchi--Hanson and multi-Taub--NUT metrics with four collinear centres, the asymptotically flat instantons on Li and Sun's manifold X_3, up to scale one for each twist at infinity in (0,1) for a fixed orientation of the torus, and, up to scale, a single further instanton with an ALF end, which is not hyperkähler. The last two are Hermitian for neither orientation. In particular, the toric asymptotically flat instanton on X_3 with a given twist is unique up to scale, as conjectured by Li and Sun. The proof combines the solitonic form of Geroch's conjecture, which turns smoothness into polynomial equations for the soliton data, with a convexity principle adapted from Aksteiner. At a fixed asymptotic parameter, the cone errors of the conical metrics with a given rod structure are the gradient of a strictly convex function of the rod lengths, and the strict convexity gives uniqueness. Existence follows from an explicit solution for the ALF end and, for the AF ends, from Li and Sun's existence theorem. We also outline how a chart of Aksteiner identifies these AF instantons with those found by Teo.
ER -