Nodal sets of Laplace eigenfunctions under small perturbations

We study the stability properties of nodal sets of Laplace eigenfunctions on compact manifolds under specific small perturbations. We prove that nodal sets are fairly stable if said perturbations are relatively small, more formally, supported at a sub-wavelength scale. We do not need any assumption on the topology of the nodal sets. As an indirect application, we are able to show that a certain "Payne property" concerning the second nodal line remains stable under controlled perturbations of the domain.