PreprintCC BY-NC-ND 4.0arXiv
Maximisers for the second Neumann eigenvalue of annuli in two-dimensional space forms
Abstract
We prove a sharp Szegö-Weinberger-type inequality for balanced doubly connected domains in 𝕊2, ℝ2 and ℍ2. More precisely, the first positive Neumann eigenvalue is bounded above by that of the corresponding rotationally symmetric annulus, determined by the area of the domain and the area of the smallest component of the complement. Equality holds only for the rotationally symmetric annulus. We also show that the balancing assumption is essential: if we drop it, the inequality fails in general.
Related
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- Minimum-Area Enclosing Rectangles and the First Neumann Eigenvalue of Planar Convex Domains
BibTeX
@misc{provenzano2026maximisers,
title = {{Maximisers for the second Neumann eigenvalue of annuli in two-dimensional space forms}},
author = {Luigi Provenzano and Alessandro Savo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10119},
eprint = {2610.10119},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10119},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Maximisers for the second Neumann eigenvalue of annuli in two-dimensional space forms
AU - Luigi Provenzano
AU - Alessandro Savo
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10119
UR - https://arxiv.org/abs/2610.10119
AB - We prove a sharp Szegö-Weinberger-type inequality for balanced doubly connected domains in 𝕊^2, ℝ^2 and ℍ^2. More precisely, the first positive Neumann eigenvalue is bounded above by that of the corresponding rotationally symmetric annulus, determined by the area of the domain and the area of the smallest component of the complement. Equality holds only for the rotationally symmetric annulus. We also show that the balancing assumption is essential: if we drop it, the inequality fails in general.
ER -