Explicit fundamental gap estimates for some convex domains in \mathbb H²

2019/11/28 by Theodora Bourni, Bourni, Theodora, Julie Clutterbuck +9
Computer Science · Mathematics · #35P15 #53C35 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Analytic and geometric function theory #Differential Geometry (math.DG) #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1911.12892

openalex publication_date 2019/11/28 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28

Abstract

Motivated by an example of Shih, we compute the fundamental gap of a family of convex domains in the hyperbolic plane \mathbb H², showing that for some of them λ₂ - λ₁ < (3π²)/(D²), where D is the diameter of the domain and λ₁, λ₂ are the first and second Dirichlet eigenvalues of the Laplace operator on the domain. The result contrasts with what is known in \mathbb Rⁿ or \mathbb Sⁿ, where λ₂ - λ₁ ≥ (3 π²)/(D²) for convex domains. We also show that the fundamental gap of the example in Shih's article is still greater than \tfrac 32 (π²)/(D²), even though the first eigenfunction of the Laplace operator is not log-concave.

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