Vol. 8, No. 1, 2015

Download this article
Download this article For screen
For printing
Recent Issues

Volume 16
Issue 7, 1485–1744
Issue 6, 1289–1483
Issue 5, 1089–1288
Issue 4, 891–1088
Issue 3, 613–890
Issue 2, 309–612
Issue 1, 1–308

Volume 15, 8 issues

Volume 14, 8 issues

Volume 13, 8 issues

Volume 12, 8 issues

Volume 11, 8 issues

Volume 10, 8 issues

Volume 9, 8 issues

Volume 8, 8 issues

Volume 7, 8 issues

Volume 6, 8 issues

Volume 5, 5 issues

Volume 4, 5 issues

Volume 3, 4 issues

Volume 2, 3 issues

Volume 1, 3 issues

The Journal
About the Journal
Editorial Board
Editors’ Interests
Submission Guidelines
Submission Form
Policies for Authors
Ethics Statement
ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Author Index
To Appear
Other MSP Journals
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