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arXiv · 2606.12745

A Large-Diameter Fundamental-Gap Lower Bound for Horoconvex Domains

Xianzhe Dai·John Ennis·Xuan Hien Nguyen·Guofang Wei
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Abstract

We prove a large-diameter fundamental-gap lower bound for compact horoconvex domains in real hyperbolic space of curvature \(-1\). The geometric part reduces large horoconvex domains to a fixed-width radial-height problem in all dimensions. The analytic part proves the needed radial-height theorem by comparing the low-energy Dirichlet form with a limiting angular operator on the sphere, while the radial complement is separated by a one-dimensional branch gap and endpoint Green estimates. The result gives the polynomial \(D^{-3}\) scale matching the Nguyen--Stancu--Wei large-diameter upper bound.

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Xianzhe Dai, John Ennis, Xuan Hien Nguyen, Guofang Wei. 2026-06-10. A Large-Diameter Fundamental-Gap Lower Bound for Horoconvex Domains. https://arxiv.org/abs/2606.12745

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