Concentration of eigenfunctions of the Laplacian on a closed Riemannian manifold

Funano, Kei; Sakurai, Yohei

doi:10.1090/proc/14430

Concentration of eigenfunctions of the Laplacian on a closed Riemannian manifold
HTML articles powered by AMS MathViewer

by Kei Funano and Yohei Sakurai

Proc. Amer. Math. Soc. 147 (2019), 3155-3164

DOI: https://doi.org/10.1090/proc/14430

Published electronically: March 5, 2019

PDF | Request permission

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.

References

Jochen Brüning, Über Knoten von Eigenfunktionen des Laplace-Beltrami-Operators, Math. Z. 158 (1978), no. 1, 15–21 (German). MR 478247, DOI 10.1007/BF01214561
J. Brüning, Nodal sets in mathematical physics, Eur. Phys. J. Special Topics 145 (2007), 181–189.
Isaac Chavel, Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, vol. 115, Academic Press, Inc., Orlando, FL, 1984. Including a chapter by Burton Randol; With an appendix by Jozef Dodziuk. MR 768584
Tobias H. Colding and William P. Minicozzi II, Volumes for eigensections, Geom. Dedicata 102 (2003), 19–24. MR 2026833, DOI 10.1023/B:GEOM.0000006578.85728.cf
Tobias H. Colding and William P. Minicozzi II, Lower bounds for nodal sets of eigenfunctions, Comm. Math. Phys. 306 (2011), no. 3, 777–784. MR 2825508, DOI 10.1007/s00220-011-1225-x
R. Courant and D. Hilbert, Methods of mathematical physics. Vol. I, Interscience Publishers, Inc., New York, N.Y., 1953. MR 0065391
Harold Donnelly and Charles Fefferman, Nodal sets of eigenfunctions on Riemannian manifolds, Invent. Math. 93 (1988), no. 1, 161–183. MR 943927, DOI 10.1007/BF01393691
B. Georgiev and M. Mukherjee, Some remarks on nodal geometry in the smooth setting, preprint, available at arXiv:1608.05344.
M. Gromov and V. D. Milman, A topological application of the isoperimetric inequality, Amer. J. Math. 105 (1983), no. 4, 843–854. MR 708367, DOI 10.2307/2374298
Q. Han and F. H. Lin, Nodal sets of solutions of Elliptic Differential Equations, book in preparation.
Dmitry Jakobson and Dan Mangoubi, Tubular neighborhoods of nodal sets and Diophantine approximation, Amer. J. Math. 131 (2009), no. 4, 1109–1135. MR 2543924, DOI 10.1353/ajm.0.0066
Michel Ledoux, The concentration of measure phenomenon, Mathematical Surveys and Monographs, vol. 89, American Mathematical Society, Providence, RI, 2001. MR 1849347, DOI 10.1090/surv/089
Alexander Logunov, Nodal sets of Laplace eigenfunctions: proof of Nadirashvili’s conjecture and of the lower bound in Yau’s conjecture, Ann. of Math. (2) 187 (2018), no. 1, 241–262. MR 3739232, DOI 10.4007/annals.2018.187.1.5
Alexander Logunov, Nodal sets of Laplace eigenfunctions: polynomial upper estimates of the Hausdorff measure, Ann. of Math. (2) 187 (2018), no. 1, 221–239. MR 3739231, DOI 10.4007/annals.2018.187.1.4
Alexander Logunov and Eugenia Malinnikova, Nodal sets of Laplace eigenfunctions: estimates of the Hausdorff measure in dimensions two and three, 50 years with Hardy spaces, Oper. Theory Adv. Appl., vol. 261, Birkhäuser/Springer, Cham, 2018, pp. 333–344. MR 3792104
Dan Mangoubi, On the inner radius of a nodal domain, Canad. Math. Bull. 51 (2008), no. 2, 249–260. MR 2414212, DOI 10.4153/CMB-2008-026-2
Y. Sakurai, Concentration of 1-Lipschitz functions on manifolds with boundary with Dirichlet boundary condition, preprint, arXiv:1712.04212.
Christopher D. Sogge, Concerning the $L^p$ norm of spectral clusters for second-order elliptic operators on compact manifolds, J. Funct. Anal. 77 (1988), no. 1, 123–138. MR 930395, DOI 10.1016/0022-1236(88)90081-X
Steve Zelditch, Local and global analysis of eigenfunctions on Riemannian manifolds, Handbook of geometric analysis. No. 1, Adv. Lect. Math. (ALM), vol. 7, Int. Press, Somerville, MA, 2008, pp. 545–658. MR 2483375
Steve Zelditch, Eigenfunctions of the Laplacian on a Riemannian manifold, CBMS Regional Conference Series in Mathematics, vol. 125, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2017. MR 3729409