Einstein link spectra and the Morse index of Ricci-flat cones
Abstract
We study the low transverse-traceless spectrum of positive Einstein manifolds and its effect on the Morse index of Ricci-flat cones. Free quaternionic quotients of four equal round spheres have exactly determined first scalar and tensor eigenvalues. For four round two-spheres, the quotient cone is nine-dimensional, is stable and has restricted holonomy SO(9), whereas its eightfold cover has infinite negative index. For a general Einstein link, its invariant eigenspaces below the Hardy threshold determine the exact index on cone annuli and the leading logarithmic accumulation of negative eigenvalues on complete Ricci-flat manifolds with weighted C2 convergence to the cone. No convergence rate is required for the leading term. At the threshold, an error of order o((log r)−2) makes cone stability equivalent to finite index. A pointwise comparison for canonically conformally Kähler Einstein four-manifolds gives, for the Chen–LeBrun–Weber metric at Einstein constant $3$, the inequality 207/100<sup Dk(s2)<21/10, where g=s−2k, s=Scalk>0 and Dk=divk∇. This proves the Hall–Haslhofer–Siepmann derivative inequality and improves the low tensor eigenvalue bound. Applications include an explicit leading coefficient for the complete Ricci-flat Böhm metrics, an index formula on long Einstein necks, and Euclidean rigidity for finite-index five-dimensional fillings under a uniform half-Weyl eigenvalue inequality.
Related
- Dirichlet deformations, Morse index, and nonlinear symmetry of Ricci-flat Böhm metrics
- Scalar curvature bounds and tangent cones for Kähler--Ricci shrinkers
- Extremal Kähler--Ricci solitons on Fano manifolds are Kähler--Einstein
- Finite-Time Singularities of the Kähler--Ricci Flow on a ℂPm-Bundle over a Product of Kähler--Einstein Manifolds
- Finite Morse index solutions of the Allen-Cahn equation with energy bound
BibTeX
@misc{nifa2026einstein,
title = {{Einstein link spectra and the Morse index of Ricci-flat cones}},
author = {Anass Nifa},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.5281/zenodo.22914003},
url = {https://hal.science/hal-05769876v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Einstein link spectra and the Morse index of Ricci-flat cones
AU - Anass Nifa
PY - 2026
DA - 2026/09/03
PB - HAL
DO - 10.5281/zenodo.22914003
UR - https://hal.science/hal-05769876v1
AB - We study the low transverse-traceless spectrum of positive Einstein manifolds and its effect on the Morse index of Ricci-flat cones. Free quaternionic quotients of four equal round spheres have exactly determined first scalar and tensor eigenvalues. For four round two-spheres, the quotient cone is nine-dimensional, is stable and has restricted holonomy SO(9), whereas its eightfold cover has infinite negative index. For a general Einstein link, its invariant eigenspaces below the Hardy threshold determine the exact index on cone annuli and the leading logarithmic accumulation of negative eigenvalues on complete Ricci-flat manifolds with weighted C^2 convergence to the cone. No convergence rate is required for the leading term. At the threshold, an error of order o((log r)^−2) makes cone stability equivalent to finite index. A pointwise comparison for canonically conformally Kähler Einstein four-manifolds gives, for the Chen–LeBrun–Weber metric at Einstein constant $3$, the inequality 207/100<sup D_k(s^2)<21/10, where g=s^−2k, s=Scal_k>0 and D_k=div_k∇. This proves the Hall–Haslhofer–Siepmann derivative inequality and improves the low tensor eigenvalue bound. Applications include an explicit leading coefficient for the complete Ricci-flat Böhm metrics, an index formula on long Einstein necks, and Euclidean rigidity for finite-index five-dimensional fillings under a uniform half-Weyl eigenvalue inequality.
ER -