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Concentration of eigenfunctions of the Laplacian on a closed Riemannian manifold


Authors: Kei Funano and Yohei Sakurai
Journal: Proc. Amer. Math. Soc. 147 (2019), 3155-3164
MSC (2010): Primary 53C21, 53C23
DOI: https://doi.org/10.1090/proc/14430
Published electronically: March 5, 2019
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Abstract: We study concentration phenomena of eigenfunctions of the Laplacian on closed Riemannian manifolds. We prove that the volume measure of a closed manifold concentrates around nodal sets of eigenfunctions exponentially. Applying the method of Colding and Minicozzi we also prove restricted exponential concentration inequalities and restricted Sogge-type $ L_p$ moment estimates of eigenfunctions.


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Kei Funano
Affiliation: Division of Mathematics & Research Center for Pure and Applied Mathematics, Graduate School of Information Sciences, Tohoku University, 6-3-09 Aramaki-Aza-Aoba, Aoba-ku, Sendai 980-8579, Japan
Email: kfunano@tohoku.ac.jp

Yohei Sakurai
Affiliation: Advanced Institute for Materials Research, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai, 980-8577, Japan
Email: yohei.sakurai.e2@tohoku.ac.jp

DOI: https://doi.org/10.1090/proc/14430
Keywords: Concentration, eigenfunctions, nodal set, Ricci curvature
Received by editor(s): December 25, 2017
Received by editor(s) in revised form: October 11, 2018, and October 13, 2018
Published electronically: March 5, 2019
Communicated by: Jia-Ping Wang
Article copyright: © Copyright 2019 American Mathematical Society