JournalsjstVol. 7, No. 1pp. 111–136

Superlevel sets and nodal extrema of Laplace–Beltrami eigenfunctions

  • Guillaume Poliquin

    Université de Montréal, Canada
Superlevel sets and nodal extrema of Laplace–Beltrami eigenfunctions cover
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Abstract

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.

Cite this article

Guillaume Poliquin, Superlevel sets and nodal extrema of Laplace–Beltrami eigenfunctions. J. Spectr. Theory 7 (2017), no. 1, pp. 111–136

DOI 10.4171/JST/157