Sign and area in nodal geometry of Laplace eigenfunctions

Abstract

The paper deals with asymptotic nodal geometry for the Laplace-Beltrami operator on closed surfaces. Given an eigenfunction f 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.