Consider a family of complex curves C→D such that the central fibre is a nodal Riemann surface while other fibres are smooth Riemann surfaces. We choose a family of conformal metrics by restricting a smooth metric on C. So near the nodes (with local models xy=t, where t is the coordinate on D), the metric is roughly the restriction of the Euclidean metric on C2={(x,y)}. Let Δt be the Laplace-Beltrami operator on the fibre over t. What is the behavior of the spectra of Δt, or rather the few lowest positive eigenvalues of Δt, as t→0?

I (sort of) know in the case of a single separating nodes, the lowest positive eigenvalue of Δt converges to 0 in a rate comparable to (log|t|)−1, while other positive eigenvalues stay uniformly away from 0. I want to know about the case that the node is non-separating, as well as the case of multiple nodes.

The result I look for is not simply the convergence of spectra, but also the rate of convergence of the few lowest eigenvalues.

More generally, is there any reference/results about the behavior of the Laplacian spectra of families of higher dimensional manifolds with normal crossing degenerations?

**Answer**

**Attribution***Source : Link , Question Author : Guangbo Xu , Answer Author : Community*