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A Sharp Diameter-Dependent Lower Bound for the First Nonzero Neumann Eigenvalue of Geodesic Triangles in Space Forms

Year: 2026

Type: preprint

Abstract: We prove a sharp lower bound for the first nonzero Neumann eigenvalue of geodesic triangles of given diameter in two-dimensional space forms. The bound is given by the first positive radial Neumann eigenvalue of an one dimensional model; it is approached by degenerating isosceles triangles. When \(K>0\) and \(D=π/(2\sqrt K)\), equality is attained precisely by birectangular triangles. We also prove a hot-spots theorem for non-acute spherical triangles of diameter at most \(π/2\), and establish antisymmetry and eigenvalue monotonicity for isosceles spherical triangles of diameter \(π/2\).

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.2608.16136

Authors: Seto, Shoo, Wei, Guofang, Xia, Yusen

Institutions: California State University, Fullerton, University of California, Santa Barbara

Topic: Spectral Theory in Mathematical Physics

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funder: National Science Foundation

Award: Fundamental Gap Estimates and Geometry /Topology of Ricci Limit Spaces

OpenAlex ID: W7203672178

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