Scalar and tensor eigenvalue bounds for conformally Kähler Einstein four-manifolds

We prove scalar and tensor eigenvalue estimates for conformally Kähler Einstein four-manifolds. For the Chen–LeBrun–Weber metric, Ric_g = Λg, the first positive scalar eigenvalue satisfies 4Λ/3 < λ₁^{sc}(g) < 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⁻ > 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. Zenodo publication date: 1 September 2026. DOI: https://doi.org/10.5281/zenodo.22851442 Author page: https://anassnifa.com/papers/scalar-tensor-eigenvalues/