Some remarks on nodal geometry in the smooth setting

TL;DR

This paper investigates the geometric properties of nodal sets and domains of Laplace eigenfunctions on smooth manifolds, providing bounds, asymptotic behaviors, and geometric insights extending previous analytic results.

Contribution

It introduces new bounds on the volume of tubular neighborhoods around nodal sets and explores geometric features of nodal domains, extending prior work to smooth manifolds using recent techniques.

Findings

Bounds on the volume of tubular neighborhoods around nodal sets

Asymptotic geometric properties of nodal domains

Quantitative estimates for inscribed ball radii in certain nodal domains

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.