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 $\left(M,g\right)$, the $\left(n-1\right)$-dimensional Hausdorff measure $|{Z}_{\lambda }|$ of the zero-set ${Z}_{\lambda }$ of an eigenfunction ${e}_{\lambda }$ of the Laplacian having eigenvalue $-\lambda$, where $\lambda \ge 1$, and normalized by ${\int }_{M}|{e}_{\lambda }{|}^{2}d{V}_{g}=1$ satisfies

$C|{Z}_{\lambda }|\ge {\lambda }^{\frac{1}{2}}{\left(\phantom{\rule{0.3em}{0ex}}{\int }_{M}|{e}_{\lambda }|\phantom{\rule{0.3em}{0ex}}d{V}_{g}\phantom{\rule{0.3em}{0ex}}\right)}^{2}$

for some uniform constant $C$. As a consequence, we recover the lower bound $|{Z}_{\lambda }|\gtrsim {\lambda }^{\left(3-n\right)∕4}$.

Keywords
eigenfunctions, nodal lines
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