Minimizers of Laplace eigenvalues under a lower curvature bound

We prove a sharp comparison, with Obata-type rigidity, for all Neumann eigenvalues of one-dimensional $\mathrm{CD}(1,2)$ spaces against the Legendre model, under a convexity condition on the density. It follows that minimizers of the $k$-th Laplace eigenvalue among closed surfaces of Gaussian curvat