PreprintarXiv licencearXiv
Interlacing of the Volume and Half-Volume Spectra
Abstract
We give a direct proof of the weak interlacing inequalities ωp≤ω̃p≤ωp+1 between the volume and half-volume spectra of a closed connected Riemannian manifold Mn+1, n+1≥3. We also prove that, for a residual set of smooth metrics, the area of every smooth embedded half-volume CMC boundary lies outside the rational span of the areas of smooth closed embedded minimal hypersurfaces. For dimensions 3≤ n+1≤ 5, these results yield strict interlacing for generic metrics and a lower bound of order An+1 for the number of distinct half-volume CMC boundaries with area at most A.
Related
- Spectra, Growth, and Localization in Half-Bounded Tensor Triangular Geometry
- Real-rootedness of the Z-polynomials of sparse paving matroids and strict interlacing for uniform matroids
- Zeros and interlacing for multiset Eulerian-Narayana polynomials
- Interlacing on the Unit Circle via Coefficientwise Reciprocals
- A proof of the interlacing conjecture for the singular values of Sylvester-Kac matrices
BibTeX
@misc{jiao2026interlacing,
title = {{Interlacing of the Volume and Half-Volume Spectra}},
author = {Xiaoxiang Jiao and Hangyue Zhu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07801},
eprint = {2611.07801},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07801},
}
RIS
TY - UNPB
TI - Interlacing of the Volume and Half-Volume Spectra
AU - Xiaoxiang Jiao
AU - Hangyue Zhu
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.07801
UR - https://arxiv.org/abs/2611.07801
AB - We give a direct proof of the weak interlacing inequalities ω_p≤ω̃_p≤ω_p+1 between the volume and half-volume spectra of a closed connected Riemannian manifold M^n+1, n+1≥3. We also prove that, for a residual set of smooth metrics, the area of every smooth embedded half-volume CMC boundary lies outside the rational span of the areas of smooth closed embedded minimal hypersurfaces. For dimensions 3≤ n+1≤ 5, these results yield strict interlacing for generic metrics and a lower bound of order A^n+1 for the number of distinct half-volume CMC boundaries with area at most A.
ER -