Coupled reciprocal-sum inequalities for Neumann eigenvalues in rank-one symmetric spaces
Abstract
Let M be a rank-one symmetric space of compact or noncompact type and real dimension m=kn, where k∈{1,2,4,8} is the real dimension of the underlying division algebra. Meng and Wang proved sharp isoperimetric inequalities for reciprocal sums of the first p or ℓ nonzero Neumann eigenvalues, with indices strictly below m in the nonconstant-curvature cases. We keep the whole m-dimensional transplanted first eigenspace of an equimeasurable geodesic ball and treat its mass and stiffness forms as a coupled generalized eigenvalue problem. For every bounded connected smooth domain in a noncompact rank-one symmetricspace we prove the full sharp estimate ∑j=1m1/μj(Ω) ≥ m/μ1(B), where B is a geodesic ball with |B|=|Ω|; equality characterizes geodesic balls. The Hopf-direction contribution is encoded by a positive, unital, trace-preserving projection on symmetric matrices, and the finite-dimensional closure is supplied by a positive-projection Jensen inequality. In compact type, under the localization hypothesis of Meng--Wang, the same full estimate is obtained whenever cot2 R≥ k−1, where R is the radius of the comparison ball, as well as in the isotropic endpoint cases and for Hopf-balanced angular second moment. Consequently the complete m-term inequality holds for all complex and quaternionic projective spaces under the original localization assumption, and for the Cayley plane when R≤ arctan(1/√7). In the remaining admissible Cayley interval we exhibit a finite-dimensional obstruction showing that the affine tangent-matrix bounds alone cannot yield the full trace estimate. This obstruction concerns the proof mechanism and is not a counterexample to the unresolved spectral inequality.
Related
- A proof of the Ashbaugh--Benguria conjecture for reciprocal sums of Neumann eigenvalues
- Inequalities between Dirichlet and Neumann eigenvalues in large dimensions
- New Qualitative and Quantitative Estimates for Neumann Eigenvalues toward Pólya's Conjecture
- A quadratic comparison of Neumann eigenvalues on thin convex domains in arbitrary dimenstion
- Neumann eigenvalues of isosceles triangles: monotonicity and asymptotics
BibTeX
@misc{meng2026coupled,
title = {{Coupled reciprocal-sum inequalities for Neumann eigenvalues in rank-one symmetric spaces}},
author = {Yifeng Meng},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19837},
eprint = {2610.19837},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19837},
note = {CC0},
}
RIS
TY - UNPB
TI - Coupled reciprocal-sum inequalities for Neumann eigenvalues in rank-one symmetric spaces
AU - Yifeng Meng
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.19837
UR - https://arxiv.org/abs/2610.19837
AB - Let M be a rank-one symmetric space of compact or noncompact type and real dimension m=kn, where k∈{1,2,4,8} is the real dimension of the underlying division algebra. Meng and Wang proved sharp isoperimetric inequalities for reciprocal sums of the first p or ℓ nonzero Neumann eigenvalues, with indices strictly below m in the nonconstant-curvature cases. We keep the whole m-dimensional transplanted first eigenspace of an equimeasurable geodesic ball and treat its mass and stiffness forms as a coupled generalized eigenvalue problem. For every bounded connected smooth domain in a noncompact rank-one symmetricspace we prove the full sharp estimate ∑_j=1^m1/μ_j(Ω) ≥ m/μ_1(B), where B is a geodesic ball with |B|=|Ω|; equality characterizes geodesic balls. The Hopf-direction contribution is encoded by a positive, unital, trace-preserving projection on symmetric matrices, and the finite-dimensional closure is supplied by a positive-projection Jensen inequality. In compact type, under the localization hypothesis of Meng--Wang, the same full estimate is obtained whenever cot^2 R≥ k−1, where R is the radius of the comparison ball, as well as in the isotropic endpoint cases and for Hopf-balanced angular second moment. Consequently the complete m-term inequality holds for all complex and quaternionic projective spaces under the original localization assumption, and for the Cayley plane when R≤ arctan(1/√7). In the remaining admissible Cayley interval we exhibit a finite-dimensional obstruction showing that the affine tangent-matrix bounds alone cannot yield the full trace estimate. This obstruction concerns the proof mechanism and is not a counterexample to the unresolved spectral inequality.
ER -