Vol. 5, No. 5, 2012

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

Volume 10
Issue 2, 253–512
Issue 1, 1–252

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 Cover
Editorial Board
Editors’ Interests
About the Journal
Scientific Advantages
Submission Guidelines
Submission Form
Editorial Login
Author Index
To Appear
ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
A natural lower bound for the size of nodal sets

Hamid Hezari and Christopher D. Sogge

Vol. 5 (2012), No. 5, 1133–1137

We prove that, for an n-dimensional compact Riemannian manifold (M,g), the (n 1)-dimensional Hausdorff measure |Zλ| of the zero-set Zλ of an eigenfunction eλ of the Laplacian having eigenvalue λ, where λ 1, and normalized by M|eλ|2dV g = 1 satisfies

C|Zλ| λ1 2 M|eλ|dV g 2

for some uniform constant C. As a consequence, we recover the lower bound |Zλ| λ(3n)4.

eigenfunctions, nodal lines
Mathematical Subject Classification 2010
Primary: 35P15
Received: 12 August 2011
Accepted: 24 October 2011
Published: 29 December 2012
Hamid Hezari
Department of Mathematics
University of California
Irvine, CA 92697
United States
Christopher D. Sogge
Department of Mathematics
Johns Hopkins University
Baltimore, MD 21093
United States