Degeneration of Riemann surfaces and small eigenvalues of the Laplacian

Abstract: For a one-parameter degeneration of compact Riemann surfaces endowed with the K\"ahler metric induced from the K\"ahler metric on the total space of the family, we determine the exact magnitude of the small eigenvalues of the Laplacian as a function on the parameter space, under the assumption that the singular fiber is reduced. The novelty in our approach is that we compute the asymptotic behavior of certain difference of (logarithm of) analytic torsions in the degeneration in two ways. On the one hand, via heat kernel estimates, it is shown that the leading asymptotic is determined by the product of the small eigenvalues. On the other hand, using Quillen metrics, the leading asymptotic is connected with the period integrals, which we explicitly evaluate.