Instability of Einstein warped products and a one-sided Weyl curvature gap
Abstract
Let Ric_g = Λg, where Λ > 0. For Einstein warped products of closed manifolds, we prove that the least transverse-traceless Einstein eigenvalue is strictly less than −(n−2)Λ/(n−1), without an upper bound on the dimension. The same estimate holds for the smooth spherical-fibre completions specified here, unless the completed metric is round. A trace-free Hessian correction expresses the shifted quadratic form as a negative sum of squared base-curvature tensors and admits an H¹ extension across the collapsing fibres. In dimension four, there is a universal positive gap for ‖(λ_max(W⁺)−2Λ/3)_+‖_{L²} on closed simply connected oriented positive Einstein manifolds. Below this gap the excess vanishes, and the metric is anti-self-dual or Kähler–Einstein in the given orientation. The proof uses Einstein orbifold compactness, extension of parallel real line bundles through anti-self-dual ALE limits, and LeBrun–Ozuch’s Kähler–Einstein desingularization theorem. For circle-warped four-metrics, the corrected quadratic form equals a negative multiple of either chiral Weyl energy. The small-excess TT-semistable metrics are the round sphere and the Fubini–Study plane. Preprint. MSC 2020: 53C25, 58J50, 53C23.Author manuscript: https://anassnifa.com/papers/einstein-instability-weyl-gap
Related
- Multiply warped products with harmonic Weyl tensor and zero radial Weyl curvature
- The Faraday Form of Conformal Products: Weyl Curvature and Four-Dimensional Einstein Rigidity
- Doubly warped product Einstein metrics on spheres
- Sharp upper bounds for Steklov eigenvalues on warped products
- Computing Smooth Geodesics under Two-Sided Curvature Bounds with Applications to Robotics and Image Analysis
BibTeX
@misc{nifa2026instability,
title = {{Instability of Einstein warped products and a one-sided Weyl curvature gap}},
author = {Anass Nifa},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.5281/zenodo.22851382},
url = {https://hal.science/hal-05769881v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Instability of Einstein warped products and a one-sided Weyl curvature gap
AU - Anass Nifa
PY - 2026
DA - 2026/09/01
PB - HAL
DO - 10.5281/zenodo.22851382
UR - https://hal.science/hal-05769881v1
AB - Let Ric_g = Λg, where Λ > 0. For Einstein warped products of closed manifolds, we prove that the least transverse-traceless Einstein eigenvalue is strictly less than −(n−2)Λ/(n−1), without an upper bound on the dimension. The same estimate holds for the smooth spherical-fibre completions specified here, unless the completed metric is round. A trace-free Hessian correction expresses the shifted quadratic form as a negative sum of squared base-curvature tensors and admits an H¹ extension across the collapsing fibres. In dimension four, there is a universal positive gap for ‖(λ_max(W⁺)−2Λ/3)_+‖_{L²} on closed simply connected oriented positive Einstein manifolds. Below this gap the excess vanishes, and the metric is anti-self-dual or Kähler–Einstein in the given orientation. The proof uses Einstein orbifold compactness, extension of parallel real line bundles through anti-self-dual ALE limits, and LeBrun–Ozuch’s Kähler–Einstein desingularization theorem. For circle-warped four-metrics, the corrected quadratic form equals a negative multiple of either chiral Weyl energy. The small-excess TT-semistable metrics are the round sphere and the Fubini–Study plane. Preprint. MSC 2020: 53C25, 58J50, 53C23.Author manuscript: https://anassnifa.com/papers/einstein-instability-weyl-gap
ER -