Some remarks on nodal geometry in the smooth setting

2016/08/18 by Georgiev, Bogdan, Mukherjee, Mayukh
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1608.05344

Abstract

We consider a Laplace eigenfunction φ_λ on a smooth closed Riemannian manifold, that is, satisfying -Δφ_λ= λφ_λ. 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 φ_λ. 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. \citeL1). 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 Ω_λ on which φ_λ satisfies exponentially small L^∞ bounds, we give some quantitative estimates for radii of inscribed balls.

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