Sharp shifted reciprocal sums of Neumann eigenvalues on space forms

2026/07/28 by Daguang Chen, Chengxi Yang
#math.DG

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Abstract

Let \mathbbM_κⁿ be the space form of sectional curvature κ∈\-1,0,1\, so that \mathbbM_-1ⁿ=ℍⁿ, \mathbbM₀ⁿ=ℝⁿ, \mathbbM₁ⁿ=\mathbbSⁿ. Let Ω⊂\mathbbM_κⁿ be a nonempty bounded open set with Lipschitz boundary, and assume that 0<|Ω|<|\mathbbSⁿ| when κ=1. Write 0=μ₀(Ω)≤μ₁(Ω)≤⋯ for the Neumann spectrum, and let B_R^κ⊂\mathbbM_κⁿ be a geodesic ball of volume \vertΩ\vert/2. We prove the sharp shifted reciprocal inequality ∑_j=2ⁿ⁺¹\frac1μ_j(Ω) ≥ (n)/(μ₁(B_R^κ)) = (n)/(μ₂(B_R^κ\sqcup B_R^κ)). Equality holds if and only if Ω is the disjoint union of two equal geodesic balls. This gives an affirmative answer to the conjecture of \cite[Remark~11]BucurMartinetNahon2025.

Citations

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