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Some remarks on nodal geometry in the smooth setting

: Georgiev, B.; Mukherjee, M.


Calculus of variations and partial differential equations 58 (2019), No.3, Art. 93, 25 pp.
ISSN: 0944-2669 (Print)
ISSN: 1432-0835 (Online)
Journal Article
Fraunhofer IAIS ()

We consider a Laplace eigenfunction φλ on a smooth closed Riemannian manifold, that is, satisfying −Δφλ=λφλ. 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 φλ. 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 Ωλ on which φλ satisfies exponentially small L∞ bounds, we give some quantitative estimates for radii of inscribed balls.