Sharp shifted reciprocal sums of Neumann eigenvalues on space forms
Let $\mathbb{M}_κ^n$ be the space form of sectional curvature $κ\in\{-1,0,1\}$, so that $\mathbb{M}_{-1}^n=\mathbb{H}^n, \mathbb{M}_{0}^n=\mathbb{R}^n, \mathbb{M}_{1}^n=\mathbb{S}^n$. Let $Ω\subset\mathbb{M}_κ^n$ be a nonempty bounded open set with Lipschitz boundary, and assume that $0<|Ω|<|\mathbb{S}^n|$ when $κ=1$. Write $0=μ_0(Ω)\leqμ_1(Ω)\leq\cdots$ for the Neumann spectrum, and let $B_R^κ\subset\mathbb{M}_κ^n$ be a geodesic ball of volume $\vertΩ\vert/2$. We prove the sharp shifted reciprocal inequality \[ \sum_{j=2}^{n+1}\frac1{μ_j(Ω)} \geq \frac{n}{μ_1(B_R^κ)} = \frac{n}{μ_2(B_R^κ\sqcup B_R^κ)}. \] 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}.