Sign and area in nodal geometry of Laplace eigenfunctions

Fëdor Nazarov*, Leonid Polterovich, Mikhail Sodin

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

23 Scopus citations

Abstract

The paper deals with asymptotic nodal geometry for the Laplace-Beltrami operator on closed surfaces. Given an eigenfunction / corresponding to a large eigenvalue, we study local asymmetry of the distribution of sign(f) with respect to the surface area. It is measured as follows: take any disc centered at the nodal line {f = 0}, and pick at random a point in this disc. What is the probability that the function assumes a positive value at the chosen point? We show that this quantity may decay logarithmically as the eigenvalue goes to infinity, but never faster than that. In other words, only a mild local asymmetry may appear. The proof combines methods due to Donnelly-Fefferman and Nadirashvili with a new result on harmonic functions in the unit disc.

Original languageEnglish
Pages (from-to)879-910
Number of pages32
JournalAmerican Journal of Mathematics
Volume127
Issue number4
DOIs
StatePublished - Aug 2005

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