Nodal sets and growth exponents of Laplace eigenfunctions on surfaces

Roy-Fortin, Guillaume

doi:10.2140/apde.2015.8.223

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Nodal sets and growth exponents of Laplace eigenfunctions on surfaces

Guillaume Roy-Fortin

Vol. 8 (2015), No. 1, 223–255

DOI: 10.2140/apde.2015.8.223

Abstract

We prove a result, announced by F. Nazarov, L. Polterovich and M. Sodin, that exhibits a relation between the average local growth of a Laplace eigenfunction on a closed surface and the global size of its nodal set. More precisely, we provide a lower and an upper bound to the Hausdorff measure of the nodal set in terms of the expected value of the growth exponent of an eigenfunction on disks of wavelength-like radius. Combined with Yau’s conjecture, the result implies that the average local growth of an eigenfunction on such disks is bounded by constants in the semiclassical limit. We also obtain results that link the size of the nodal set to the growth of solutions of planar Schrödinger equations with small potential.

Keywords

spectral geometry, Laplace eigenfunctions, nodal sets, growth of eigenfunctions

Mathematical Subject Classification 2010

Primary: 58J50

Milestones

Received: 6 September 2014

Accepted: 26 November 2014

Published: 15 April 2015

Authors

Guillaume Roy-Fortin
Département de mathématiques et de statistique Université de Montréal CP 6128 succ. Centre-Ville Montréal, QB H3C 3J7 Canada

Vol. 8, No. 1, 2015