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- W2964248932 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." @default.
- W2964248932 created "2019-07-30" @default.
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- W2964248932 date "2019-05-08" @default.
- W2964248932 modified "2023-09-26" @default.
- W2964248932 title "Some remarks on nodal geometry in the smooth setting" @default.
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- W2964248932 doi "https://doi.org/10.1007/s00526-019-1541-0" @default.
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