Xia–Wang reciprocal Neumann eigenvalue conjecture for non-Euclidean space forms

Let $M^n_\kappa$ be the simply connected $n$-dimensional Riemannian manifold of constant sectional curvature $\kappa\in\{-1,1\}$, and let $\Omega\subset M^n_\kappa$ be a bounded connected domain with smooth boundary. Assume that, when $\kappa=1$, $\Omega$ is contained in an open hemisphere. Let $B_\Omega\subset M^n_\kappa$ be a geodesic ball with $|B_\Omega|=|\Omega|$, and let $\mu_j(\Omega)$ denote the positive Neumann eigenvalues of $\Omega$. Xia–Wang conjecture.

$$\sum_{j=1}^n\frac{1}{\mu_j(\Omega)}\geq\frac{n}{\mu_1(B_\Omega)}.$$

Equality holds if and only if $\Omega$ is isometric to $B_\Omega$. The conjecture extends the sharp reciprocal-sum problem from Euclidean domains to hyperbolic space and to spherical domains contained in an open hemisphere; the abstract states that the paper proves it, so its database status is solved.