On the effects of small perturbation on low energy Laplace eigenfunctions

Abstract

We investigate several aspects of the nodal geometry and topology of Laplace eigenfunctions, with particular emphasis on the low-frequency regime. This includes investigations in and around the well-known nodal line conjecture, opening angle estimates of nodal domains, saturation of (fundamental) spectral gaps etc., and behaviour of all of the above under small-scale perturbations. We aim to highlight interesting aspects of spectral theory and nodal phenomena tied to ground state/low energy eigenfunctions, as opposed to asymptotic results.