Sharp shifted reciprocal sums of Neumann eigenvalues on space forms

Abstract

Let ${\mathbb{M}}_{\kappa }^n$ be the space form of sectional curvature $\kappa \in \{-1,0,1\}$, so that ${\mathbb{M}}_{-1}^n={\mathbb{H}}^n,{\mathbb{M}}_0^n={\mathbb{R}}^n,{\mathbb{M}}_1^n={\mathbb{S}}^n$. Let $\Omega \subset {\mathbb{M}}_{\kappa }^n$ be a nonempty bounded open set with Lipschitz boundary, and assume that $0<|\Omega |<|{\mathbb{S}}^n|$ when $\kappa =1$. Write $0={\mu }_0(\Omega )\le {\mu }_1(\Omega )\le \cdots$ for the Neumann spectrum, and let $B_R^{\kappa }\subset {\mathbb{M}}_{\kappa }^n$ be a geodesic ball of volume $|\Omega |/2$. 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 $\Omega$ is the disjoint union of two equal geodesic balls. This gives an affirmative answer to the conjecture of \cite[Remark~11]{BucurMartinetNahon2025}.