LOCAL VERSION OF COURANT’S NODAL DOMAIN THEOREM

Sagun Chanillo, Alexander Logunov, Eugenia Malinnikova, Dan Mangoubi

Research output: Contribution to journalArticlepeer-review

Abstract

Let (Mn, g) be a compact n-dimensional Riemannian manifold without boundary, where g = (gij) is C1-smooth. Consider the sequence of eigenfunctions uk of the Laplace operator on M. Let B be a ball on M. We prove that the number of nodal domains of uk that intersect B is not greater than C1 VolumeVolumegg((MB))k + C2kn− n1 , where C1, C2 depend on M. The problem of local bounds for the volume and for the number of nodal domains was raised by Donnelly and Fefferman, who also proposed an idea how one can prove such bounds. We combine their idea with two ingredients: the recent sharp Remez type inequality for eigenfunctions and the Landis type growth lemma in narrow domains.

Original languageAmerican English
Pages (from-to)49-63
Number of pages15
JournalJournal of Differential Geometry
Volume126
Issue number1
DOIs
StatePublished - Jan 2024

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