Tracking Eigenvalues to the Frontier of Moduli Space I. Convergence and Spectral Accumulation

We consider the spectral behavior of the Laplace–Beltrami operator associated to a class of singular perturbations of a Riemannian metric on a complete manifold. The class of perturbations generalizes the well-known “opening node” perturbation of Teichmüller theory. In particular, we recover results of L. Ji and M. Zworski (J. Func. Anal.114 (1993), 412–420) and S. Wolpert (Invent. Math.108 (1992), 91–129) from our more general methods.