Scalar and tensor eigenvalue bounds for conformally Kähler Einstein four-manifolds
Abstract
We prove scalar and tensor eigenvalue estimates for conformally Kähler Einstein four-manifolds. For the Chen–LeBrun–Weber metric, Ricg=Λ g, the first positive scalar eigenvalue satisfies 4Λ/3lt λ1sc(g)lt 1951Λ/1000. Thus this metric admits a destabilizing conformal variation, as asserted in Hall–Murphy's Conjecture 5.3. The upper bound follows from boundary moments of an affine quotient on the moment polygon and a positive-coefficient polynomial identity. For any closed Einstein four-manifold conformal to a positive-scalar-curvature Kähler metric, put m=b2-gt 0 and ρ=max sk/min sk, using the complex orientation. We prove λmTT≤(4Λ/3)(1−ρ−3) for the full Lichnerowicz operator, with strict inequality for a nonconstant conformal factor. For the Chen–LeBrun–Weber metric, this gives two TT eigenvalues below 7Λ/6, proves the instability of its normalized Ricci-flat cone, and, together with the scalar bound, gives at least three positive directions for the entropy Hessian modulo diffeomorphisms and scaling. Finally, under det W+≥0, the stable cones over closed oriented positive Einstein four-manifolds are precisely those over the round sphere and the Fubini–Study projective plane.
Related
- Dirichlet eigenvalue asymptotics on conformally compact Einstein manifolds
- Locally Conformally Flat Manifolds with Positive Scalar Curvature: Kleinian Groups, Moduli Spaces, and Euclidean Rigidity
- Rigidity of Einstein Manifolds under Two Eigenvalue Conditions on the Curvature Operator of the Second Kind
- Instability of conformally Kähler, Einstein metrics
- On products of locally conformally Kähler manifolds
BibTeX
@misc{nifa2026scalar,
title = {{Scalar and tensor eigenvalue bounds for conformally Kähler Einstein four-manifolds}},
author = {Anass Nifa},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.5281/zenodo.22851442},
url = {https://hal.science/hal-05769995v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Scalar and tensor eigenvalue bounds for conformally Kähler Einstein four-manifolds
AU - Anass Nifa
PY - 2026
DA - 2026/09/01
PB - HAL
DO - 10.5281/zenodo.22851442
UR - https://hal.science/hal-05769995v1
AB - We prove scalar and tensor eigenvalue estimates for conformally Kähler Einstein four-manifolds. For the Chen–LeBrun–Weber metric, Ric_g=Λ g, the first positive scalar eigenvalue satisfies 4Λ/3lt λ_1^sc(g)lt 1951Λ/1000. Thus this metric admits a destabilizing conformal variation, as asserted in Hall–Murphy's Conjecture 5.3. The upper bound follows from boundary moments of an affine quotient on the moment polygon and a positive-coefficient polynomial identity. For any closed Einstein four-manifold conformal to a positive-scalar-curvature Kähler metric, put m=b_2^-gt 0 and ρ=max s_k/min s_k, using the complex orientation. We prove λ_m^TT≤(4Λ/3)(1−ρ^−3) for the full Lichnerowicz operator, with strict inequality for a nonconstant conformal factor. For the Chen–LeBrun–Weber metric, this gives two TT eigenvalues below 7Λ/6, proves the instability of its normalized Ricci-flat cone, and, together with the scalar bound, gives at least three positive directions for the entropy Hessian modulo diffeomorphisms and scaling. Finally, under det W^+≥0, the stable cones over closed oriented positive Einstein four-manifolds are precisely those over the round sphere and the Fubini–Study projective plane.
ER -