Sugata Mondal

Spectral instability of coverings

Randol (1974) showed that every compact Riemannian surface S has a (Riemannian) covering p: S' --> S such that the first (non-zero) eigenvalue of the Laplacian of S' is strictly below that of S. Recently, Magee et al. has shown that, in many situations, if the first (non-zero) eigenvalue of S is below a threshold, then for almost any n-sheeted cover S'' of S the first (non-zero) eigenvalue of S and S'' are the same (as n tends to infinity, with respect to uniform probability measure on the space of all n-sheeted covers of S). We call this phenomenon \lambda_1-stability for the covering S'' --> S. In this talk, I will provide a necessary condition for such stability, and deduce examples of \lambda_1-unstable coverings.