Journal of Spectral Theory

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Volume 7, Issue 1, 2017, pp. 111–136
DOI: 10.4171/JST/157

Published online: 2017-03-22

Superlevel sets and nodal extrema of Laplace–Beltrami eigenfunctions

Guillaume Poliquin[1]

(1) Université de Montréal, Canada

We estimate the volume of superlevel sets of Laplace–Beltrami eigenfunctions on a compact Riemannian manifold. The proof uses the Green’s function representation and the Bathtub principle. As an application, we obtain upper bounds on the distribution of the extrema of a Laplace–Beltrami eigenfunction over its nodal domains. Such bounds have been previously proved by L. Polterovich and M. Sodin in the case of compact surfaces. Our techniques allow to generalize these results to arbitrary dimensions. We also discuss a dierent approach to the problem based on reverse Hölder inequalities due to G. Chiti.

Keywords: Laplacian, Riemannian manifold, eigenfunction, nodal domain, bathtub principle

Poliquin Guillaume: Superlevel sets and nodal extrema of Laplace–Beltrami eigenfunctions. J. Spectr. Theory 7 (2017), 111-136. doi: 10.4171/JST/157