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. Ann Math (2) 187(1):241–262, 2018). 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.