Some remarks on nodal geometry in the smooth setting

Abstract

We consider a Laplace eigenfunction $\varphi_\lambda$ on a smooth closed Riemannian manifold, that is, satisfying $-\Delta \varphi_\lambda = \lambda \varphi_\lambda$. We introduce several observations about the geometry of its vanishing (nodal) set and corresponding nodal domains. First, we give asymptotic upper and lower bounds on the volume of a tubular neighbourhood around the nodal set of $\varphi_\lambda$. This extends previous work of Jakobson and Mangoubi in case $ (M, g) $ is real-analytic. A significant ingredient in our discussion are some recent techniques due to Logunov (cf. \cite{L1}). Second, we exhibit some remarks related to the asymptotic geometry of nodal domains. In particular, we observe an analogue of a result of Cheng in higher dimensions regarding the interior opening angle of a nodal domain at a singular point. Further, for nodal domains $\Omega_\lambda$ on which $\varphi_\lambda$ satisfies exponentially small $L^\infty$ bounds, we give some quantitative estimates for radii of inscribed balls.