Compact toric Einstein four-manifolds
Abstract
This is a continuation of our previous paper on toric gravitational instantons. We develop a systematic theory of compact simply-connected Einstein four-manifolds with toric symmetry. Applications include (1) The construction of simply connected positive Einstein four-manifolds with arbitrarily large b2. Our method is based on local gluing combined with a global virtual counting argument. The key idea involves a particular design of rod structures to enable a gluing construction and restrict possible degenerations. (2) A topological characterization of algebraically special toric Einstein metrics. This uses special curvature identities for W+ in the toric setting. (3) Uniqueness, up to scaling and isometry, on toric four-manifolds admitting algebraically special Einstein metrics. The proofs uses a variational study of the Einstein--Hilbert functional. (4) Diffeomorphism classification of compact toric Einstein four-manifolds in terms of an improved Hitchin-Thorpe inequality χ≥3|τ|. In particular, our construction recovers all the classical toric Einstein metrics, including the Page and Chen--LeBrun--Weber metrics, starting from the round sphere through moduli spaces of conical Einstein metrics.
Related
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- On Positively Curved Einstein Four-Manifolds with Euler Characteristic at Most Three
- Scalar and tensor eigenvalue bounds for conformally Kähler Einstein four-manifolds
BibTeX
@misc{li2026compact,
title = {{Compact toric Einstein four-manifolds}},
author = {Mingyang Li and Song Sun},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10425},
eprint = {2610.10425},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10425},
}
RIS
TY - UNPB
TI - Compact toric Einstein four-manifolds
AU - Mingyang Li
AU - Song Sun
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10425
UR - https://arxiv.org/abs/2610.10425
AB - This is a continuation of our previous paper on toric gravitational instantons. We develop a systematic theory of compact simply-connected Einstein four-manifolds with toric symmetry. Applications include (1) The construction of simply connected positive Einstein four-manifolds with arbitrarily large b_2. Our method is based on local gluing combined with a global virtual counting argument. The key idea involves a particular design of rod structures to enable a gluing construction and restrict possible degenerations. (2) A topological characterization of algebraically special toric Einstein metrics. This uses special curvature identities for W^+ in the toric setting. (3) Uniqueness, up to scaling and isometry, on toric four-manifolds admitting algebraically special Einstein metrics. The proofs uses a variational study of the Einstein--Hilbert functional. (4) Diffeomorphism classification of compact toric Einstein four-manifolds in terms of an improved Hitchin-Thorpe inequality χ≥3|τ|. In particular, our construction recovers all the classical toric Einstein metrics, including the Page and Chen--LeBrun--Weber metrics, starting from the round sphere through moduli spaces of conical Einstein metrics.
ER -