Spectral almost rigidity on Kähler manifolds with positive Ricci lower bound
Abstract
In this work, a sharp Kähler spectral almost-rigidity theorem was established, resolving Conjecture 1.8 of Chu--Wang--Zhang. For compact Kähler manifolds satisfying Ric(ω)≥ω, pinching the first n2+3 nonzero complex eigenvalues to one forces the manifolds to be Gromov Hausdorff close to normalized complex projective space and determines the biholomorphism type. The smaller index n2+1 is the sharp threshold for noncollapsing. More generally, if a normalized measured limit has essential real dimension r, then the multiplicity of the critical eigenvalue is at most r+⌊ r/2⌋2, with equality attained in every dimension. A uniform energy estimate for kernel projections of complex gradient Gram matrices leads to a Lie algebra action on the whole spectral resolution.
Related
- Gromov-Hausdorff instability of the positive mass theorem for Kähler manifolds
- Fundamental groups of compact Kähler manifolds with semi-positive holomorphic sectional curvature
- Spectral Improvements of Geometric Inequalities on Closed Kähler Manifolds
- BALANCING THE 2-SYSTOLES OF SOME KÄHLER MANIFOLDS WITH POSITIVE SCALAR CURVATURE
- Convergence of cohomogeneity-one Lagrangian mean curvature flow in positive Kähler-Einstein manifolds
BibTeX
@misc{wang2026spectral,
title = {{Spectral almost rigidity on Kähler manifolds with positive Ricci lower bound}},
author = {Haohao Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12525},
eprint = {2609.12525},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12525},
}
RIS
TY - UNPB
TI - Spectral almost rigidity on Kähler manifolds with positive Ricci lower bound
AU - Haohao Wang
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.12525
UR - https://arxiv.org/abs/2609.12525
AB - In this work, a sharp Kähler spectral almost-rigidity theorem was established, resolving Conjecture 1.8 of Chu--Wang--Zhang. For compact Kähler manifolds satisfying Ric(ω)≥ω, pinching the first n^2+3 nonzero complex eigenvalues to one forces the manifolds to be Gromov Hausdorff close to normalized complex projective space and determines the biholomorphism type. The smaller index n^2+1 is the sharp threshold for noncollapsing. More generally, if a normalized measured limit has essential real dimension r, then the multiplicity of the critical eigenvalue is at most r+⌊ r/2⌋^2, with equality attained in every dimension. A uniform energy estimate for kernel projections of complex gradient Gram matrices leads to a Lie algebra action on the whole spectral resolution.
ER -