Scaling inequalities and limits for clamped plate eigenvalues on geodesic disks
Abstract
For the bilaplacian in spherical and hyperbolic spaces, clamped vibrating and buckling eigenvalues on geodesic disks are shown to satisfy scaling inequalities analogous to the standard scale invariance of Euclidean plate eigenvalues. These results extend curved-space scaling results from the Laplacian to fourth-order eigenvalue problems. In addition, the limiting behavior of the scaled eigenvalues is determined as the disks expand to fill the ambient space. Depending on the geometry and the choice of scaling, the eigenvalues tend to zero, positive or negative infinity, or finite values associated with the bottom of the hyperbolic spectrum. The negative divergence in one spherical buckling limit is related by conformal inversion to an exterior buckling problem.
Related
- Principal eigenvalues of nonlocal operators with advection: sharp scaling limits and spectral phase transitions
- Sharp Harmonic-Mean Bounds for Neumann Eigenvalues on Riemannian Surfaces and Counterexamples to Geodesic Ball Optimality on Higher-Dimensional Spheres
- Inequalities between Dirichlet and Neumann eigenvalues in large dimensions
- Integral curvature estimates and geodesic limits for stable abelian Higgs fields
- Arakelov inequalities and characterization of totally geodesic ball quotients in 𝒜g
BibTeX
@misc{harman2026scaling,
title = {{Scaling inequalities and limits for clamped plate eigenvalues on geodesic disks}},
author = {Scott Harman},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29498},
eprint = {2608.29498},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29498},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Scaling inequalities and limits for clamped plate eigenvalues on geodesic disks
AU - Scott Harman
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2608.29498
UR - https://arxiv.org/abs/2608.29498
AB - For the bilaplacian in spherical and hyperbolic spaces, clamped vibrating and buckling eigenvalues on geodesic disks are shown to satisfy scaling inequalities analogous to the standard scale invariance of Euclidean plate eigenvalues. These results extend curved-space scaling results from the Laplacian to fourth-order eigenvalue problems. In addition, the limiting behavior of the scaled eigenvalues is determined as the disks expand to fill the ambient space. Depending on the geometry and the choice of scaling, the eigenvalues tend to zero, positive or negative infinity, or finite values associated with the bottom of the hyperbolic spectrum. The negative divergence in one spherical buckling limit is related by conformal inversion to an exterior buckling problem.
ER -