Vol. 5, No. 5, 2012

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A natural lower bound for the size of nodal sets

Hamid Hezari and Christopher D. Sogge

Vol. 5 (2012), No. 5, 1133–1137
Abstract

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.

Keywords
eigenfunctions, nodal lines
Mathematical Subject Classification 2010
Primary: 35P15
Milestones
Received: 12 August 2011
Accepted: 24 October 2011
Published: 29 December 2012
Authors
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