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An Improved Bound for the Ovals Problem
Abstract
Let γ⊂ℝm, m≥2, be a closed curve of length 2π with its curvature κ, parametrized by arc length, and let λγ be the first eigenvalue of the periodic curvature Schrödinger operator −d2/d s2+κ(s)2. We obtain λγ≥ √π/2 (Γ(7/6)/Γ(5/3))3. This is a near-sharp lower bound for the Ovals problem. Our proof introduces a new geometric approach. We derive a convolution identity from the closure condition and combine it with projection averaging over tangent directions and sharp Poincaré inequalities on antipodal arcs. As applications, we provide an improved two-state kinetic Lieb-Thirring inequality and the corresponding two-eigenvalue constant.
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BibTeX
@misc{suragan2026improved,
title = {{An Improved Bound for the Ovals Problem}},
author = {Durvudkhan Suragan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10775},
eprint = {2609.10775},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10775},
}
RIS
TY - UNPB
TI - An Improved Bound for the Ovals Problem
AU - Durvudkhan Suragan
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.10775
UR - https://arxiv.org/abs/2609.10775
AB - Let γ⊂ℝ^m, m≥2, be a closed curve of length 2π with its curvature κ, parametrized by arc length, and let λ_γ be the first eigenvalue of the periodic curvature Schrödinger operator −d^2/d s^2+κ(s)^2. We obtain λ_γ≥ √π/2 (Γ(7/6)/Γ(5/3))^3. This is a near-sharp lower bound for the Ovals problem. Our proof introduces a new geometric approach. We derive a convolution identity from the closure condition and combine it with projection averaging over tangent directions and sharp Poincaré inequalities on antipodal arcs. As applications, we provide an improved two-state kinetic Lieb-Thirring inequality and the corresponding two-eigenvalue constant.
ER -