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 curvature at least $1$ cannot collapse in the measured Gromov-Hausdorff completion, for every $k\ge2$. We also give a variational proof that smooth minimizers are round.
Let $(M,g)$ be a connected, compact, $n$-dimensional Riemannian manifold with $\operatorname{Ric}(M,g)\geq-(n-1)\kappa g$. We introduce a weighted combinatorial Laplacian on $\varepsilon$-discretizations of $M$ and prove a spectral comparison theorem between the weighted graph Laplacian and the Laplace-Beltrami operator. More precisely, the eigenvalues of the two operators are uniformly comparable with constants depending only on $n,\kappa,\varepsilon$, independently of the injectivity radius. As an application, we prove spectral stability under measured Gromov-Hausdorff convergence. We also recover the Schoen-Wolpert-Yau inequality using the weighted discretization on families of pinching genus-$2$ hyperbolic surfaces.
Aditya Tiwari· 0 citations
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