Positive Einstein metrics on infinitely many four-manifolds
Abstract
We construct smooth Einstein metrics with positive scalar curvature on #k(𝕊2× 𝕊2) and on kℂℙ2#(2k+1)ℂℙ̅2, for kgeqslant2. In particular, we show that positive Einstein metrics exist on infinitely many homeomorphism types of closed $4$-manifolds. The first family shows that any simply connected spin $4$-manifold carrying metrics with positive scalar curvature is homeomorphic to an Einstein $4$-manifold. Wick rotation gives associated Lorentzian static axisymmetric Einstein metrics with positive cosmological constant and arbitrarily many horizons with the same surface gravity. The metrics are obtained by varying the cone angle 2πβ of a prescribed singularity along a torus, starting from the product metric on 𝕊2×𝕊2 and from the symmetric Kähler-Einstein metric on ℂℙ2#3ℂℙ̅2. At β=2/(k+1), changing the period of the angle around the torus removes the cone singularity and changes the topology. For both families, as k→+∞, the metrics collapse along the torus orbits to an explicit $2$-dimensional disk, whose boundary has Hausdorff dimension 4/3. At the scale of curvature, they converge to complete periodic Ricci-flat metrics with Kasner asymptotics: the Euclidean Myers-Korotkin-Nicolai metric for the first family, and a new one for the second.
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BibTeX
@misc{ozuch2026positive,
title = {{Positive Einstein metrics on infinitely many four-manifolds}},
author = {Tristan Ozuch},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08052},
eprint = {2610.08052},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08052},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Positive Einstein metrics on infinitely many four-manifolds
AU - Tristan Ozuch
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08052
UR - https://arxiv.org/abs/2610.08052
AB - We construct smooth Einstein metrics with positive scalar curvature on #_k(𝕊^2× 𝕊^2) and on kℂℙ^2#(2k+1)ℂℙ̅^2, for kgeqslant2. In particular, we show that positive Einstein metrics exist on infinitely many homeomorphism types of closed $4$-manifolds. The first family shows that any simply connected spin $4$-manifold carrying metrics with positive scalar curvature is homeomorphic to an Einstein $4$-manifold. Wick rotation gives associated Lorentzian static axisymmetric Einstein metrics with positive cosmological constant and arbitrarily many horizons with the same surface gravity. The metrics are obtained by varying the cone angle 2πβ of a prescribed singularity along a torus, starting from the product metric on 𝕊^2×𝕊^2 and from the symmetric Kähler-Einstein metric on ℂℙ^2#3ℂℙ̅^2. At β=2/(k+1), changing the period of the angle around the torus removes the cone singularity and changes the topology. For both families, as k→+∞, the metrics collapse along the torus orbits to an explicit $2$-dimensional disk, whose boundary has Hausdorff dimension 4/3. At the scale of curvature, they converge to complete periodic Ricci-flat metrics with Kasner asymptotics: the Euclidean Myers-Korotkin-Nicolai metric for the first family, and a new one for the second.
ER -