Consider a family of complex curves ${\mathcal C} \to {\mathbb D}$ 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 ${\mathcal C}$. So near the nodes (with local models $xy = t$, where $t$ is the coordinate on ${\mathbb D}$), the metric is roughly the restriction of the Euclidean metric on ${\mathbb C}^2 = \{(x, y)\}$. Let $\Delta_t$ be the Laplace-Beltrami operator on the fibre over $t$. What is the behavior of the spectra of $\Delta_t$, or rather the few lowest positive eigenvalues of $\Delta_t$, as $t\to 0$?

I (sort of) know in the case of a single separating nodes, the lowest positive eigenvalue of $\Delta_t$ 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?


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