Vol. 8, No. 1, 2015

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Nodal sets and growth exponents of Laplace eigenfunctions on surfaces

Guillaume Roy-Fortin

Vol. 8 (2015), No. 1, 223–255

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.

spectral geometry, Laplace eigenfunctions, nodal sets, growth of eigenfunctions
Mathematical Subject Classification 2010
Primary: 58J50
Received: 6 September 2014
Accepted: 26 November 2014
Published: 15 April 2015
Guillaume Roy-Fortin
Département de mathématiques et de statistique
Université de Montréal
CP 6128 succ.
Montréal, QB H3C 3J7