Spectral Estimates for Compact Riemann Surfaces via Kähler Potentials
Abstract
We establish quantitative comparisons between the Laplace--Beltrami spectra of Kähler metrics of equal area on a compact Riemann surface. An estimate in terms of the oscillation of a potential gives bounds for fractional powers of reciprocal eigenvalues and explicit intervals for eigenvalue ratios. Bounds involving the gradient and Laplacian of the potential give further comparisons. On the Riemann sphere, we obtain estimates for individual eigenvalues, reciprocal sums and counting functions. For a Kähler metric on the unit 2-sphere which is centered in the sense that the unit normal vector field integrates to zero, the Dirichlet energy of the potential yields quantitative improvements of Hersch's bounds for the first positive eigenvalue and the sum of the first three reciprocal eigenvalues.
Related
- Finiteness of Fuchsian equations with unitarizable monodromy on compact Riemann surfaces
- Uniform Green packing on compact Riemann surfaces
- Exponential Green energies, quantization, and minimizers on compact Riemann surfaces
- On supersoluble like groups of automorphisms of compact Riemann surfaces
- Resolvent bounds and eigenvalue estimates of generalized Schrödinger operators with complex potentials on compact manifolds
BibTeX
@misc{liu2026spectral,
title = {{Spectral Estimates for Compact Riemann Surfaces via Kähler Potentials}},
author = {Hanwen Liu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14333},
eprint = {2609.14333},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14333},
}
RIS
TY - UNPB
TI - Spectral Estimates for Compact Riemann Surfaces via Kähler Potentials
AU - Hanwen Liu
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.14333
UR - https://arxiv.org/abs/2609.14333
AB - We establish quantitative comparisons between the Laplace--Beltrami spectra of Kähler metrics of equal area on a compact Riemann surface. An estimate in terms of the oscillation of a potential gives bounds for fractional powers of reciprocal eigenvalues and explicit intervals for eigenvalue ratios. Bounds involving the gradient and Laplacian of the potential give further comparisons. On the Riemann sphere, we obtain estimates for individual eigenvalues, reciprocal sums and counting functions. For a Kähler metric on the unit 2-sphere which is centered in the sense that the unit normal vector field integrates to zero, the Dirichlet energy of the potential yields quantitative improvements of Hersch's bounds for the first positive eigenvalue and the sum of the first three reciprocal eigenvalues.
ER -