Detecting Einstein metrics on products of spheres
Abstract
On S5× S2, S5× S3, S5× S4, S7× S2, we construct infinitely many pairwise non-isometric real-analytic Einstein metrics. After discarding finitely many members, none is isometric or homothetic to a Riemannian product or to a classical two-summand Böhm metric. The metrics have the form dt2+a(t)2gSp+b(t)2gSp+c(t)2gSq, p,q≥2, 2p+q≤8. The construction starts with solutions for which the two non-collapsing radii at one singular orbit are very small. After magnifying this region, the Einstein equations approach a Ricci-flat problem whose solutions tend to a cone by a decaying oscillation. The main point is to determine exactly which part of this oscillation can still be seen by the closing conditions at the other end of the manifold. Rather than following the full, increasingly oscillatory shooting map, we extract two response quantities from the Ricci-flat limit. One determines the common oscillatory phase; after that phase has been removed, the second closing condition is reduced to a scalar sign change. A uniform inner--outer matching argument transfers this limiting sign change to the exact Einstein equations. As the initial scale tends to zero, the phase makes arbitrarily many turns, and the closing conditions are therefore met infinitely many times.
Related
- Doubly warped product Einstein metrics on spheres
- Coupling perturbations of the conformal Einstein-Dirac system on round spheres
- Instability of conformally Kähler, Einstein metrics
- Sasaki-Einstein rational homology spheres, rational varieties and the Berglund-Hübsch rule
- Smooth manifolds homotopy equivalent to products of spheres
BibTeX
@misc{siffert2026detecting,
title = {{Detecting Einstein metrics on products of spheres}},
author = {Anna Siffert},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00066},
eprint = {2610.00066},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00066},
}
RIS
TY - UNPB
TI - Detecting Einstein metrics on products of spheres
AU - Anna Siffert
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00066
UR - https://arxiv.org/abs/2610.00066
AB - On S^5× S^2, S^5× S^3, S^5× S^4, S^7× S^2, we construct infinitely many pairwise non-isometric real-analytic Einstein metrics. After discarding finitely many members, none is isometric or homothetic to a Riemannian product or to a classical two-summand Böhm metric. The metrics have the form dt^2+a(t)^2g_S^p+b(t)^2g_S^p+c(t)^2g_S^q, p,q≥2, 2p+q≤8. The construction starts with solutions for which the two non-collapsing radii at one singular orbit are very small. After magnifying this region, the Einstein equations approach a Ricci-flat problem whose solutions tend to a cone by a decaying oscillation. The main point is to determine exactly which part of this oscillation can still be seen by the closing conditions at the other end of the manifold. Rather than following the full, increasingly oscillatory shooting map, we extract two response quantities from the Ricci-flat limit. One determines the common oscillatory phase; after that phase has been removed, the second closing condition is reduced to a scalar sign change. A uniform inner--outer matching argument transfers this limiting sign change to the exact Einstein equations. As the initial scale tends to zero, the phase makes arbitrarily many turns, and the closing conditions are therefore met infinitely many times.
ER -