Einstein metrics at conical limits: stability, disconnected moduli, and entropy rigidity
Abstract
We study three variational problems for Einstein metrics with conical limits. For every k≥5, the complete Ricci-flat Böhm metric on ℝk+1× Sk has holonomy SO(2k+1) and a homogeneous coercive estimate on all symmetric two-tensors. It attracts sufficiently small smooth L2∩ Cb2,η perturbations under the fixed-background Ricci–DeTurck flow, with convergence in L2 and every uniform Cj norm. Uniformly small ancient perturbations with bounded L2 norm are stationary. For p,q≥2 and p+q≤8, we construct distinct normalized Einstein volumes on Sp+q+1, on Sp+1× Q for every closed connected positive Einstein q-manifold Q, and on the specified identity doubles of disk bundles with vanishing O'Neill tensor. Their full Einstein moduli spaces have infinitely many connected components. The volumes approach the sine-cone value from above; specified subsequences have geometric asymptotics. On a marked conical end of rate greater than (n−2)/2, with absolutely integrable scalar curvature, a large-scale upper bound for Perelman's μ-functional has coefficient λnc−mC. No spin or scalar-curvature sign is assumed in this comparison. A parallel spinor compatible with the end induced in the universal cover gives mass–energy rigidity and characterizes equality in the global cone entropy bound by Ricci-flatness. Equality also makes the end homomorphism surjective with finite image and extends the compatible cone spinors. An ancient Ricci flow with one such slice is stationary when its finite backward Nash entropy is at least the cone value.
Related
- Exact counting of spherical metrics with one conical singularity on rectangular tori
- Instability of conformally Kähler, Einstein metrics
- CSC Bergman metrics are Einstein on pseudoconvex domains
- Sasaki-Einstein metrics on connected sums of S3× S4
- Harmonic-Einstein maps and critical metrics for eigenvalues of the conformal Laplacian
BibTeX
@misc{nifa2026einstein,
title = {{Einstein metrics at conical limits: stability, disconnected moduli, and entropy rigidity}},
author = {Anass Nifa},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.5281/zenodo.22914134},
url = {https://hal.science/hal-05769958v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Einstein metrics at conical limits: stability, disconnected moduli, and entropy rigidity
AU - Anass Nifa
PY - 2026
DA - 2026/09/18
PB - HAL
DO - 10.5281/zenodo.22914134
UR - https://hal.science/hal-05769958v1
AB - We study three variational problems for Einstein metrics with conical limits. For every k≥5, the complete Ricci-flat Böhm metric on ℝ^k+1× S^k has holonomy SO(2k+1) and a homogeneous coercive estimate on all symmetric two-tensors. It attracts sufficiently small smooth L^2∩ C_b^2,η perturbations under the fixed-background Ricci–DeTurck flow, with convergence in L^2 and every uniform C^j norm. Uniformly small ancient perturbations with bounded L^2 norm are stationary. For p,q≥2 and p+q≤8, we construct distinct normalized Einstein volumes on S^p+q+1, on S^p+1× Q for every closed connected positive Einstein q-manifold Q, and on the specified identity doubles of disk bundles with vanishing O'Neill tensor. Their full Einstein moduli spaces have infinitely many connected components. The volumes approach the sine-cone value from above; specified subsequences have geometric asymptotics. On a marked conical end of rate greater than (n−2)/2, with absolutely integrable scalar curvature, a large-scale upper bound for Perelman's μ-functional has coefficient λ_nc−m_C. No spin or scalar-curvature sign is assumed in this comparison. A parallel spinor compatible with the end induced in the universal cover gives mass–energy rigidity and characterizes equality in the global cone entropy bound by Ricci-flatness. Equality also makes the end homomorphism surjective with finite image and extends the compatible cone spinors. An ancient Ricci flow with one such slice is stationary when its finite backward Nash entropy is at least the cone value.
ER -