Problems related to the concentration of eigenfunctions
Journées équations aux dérivées partielles (2015), article no. 9, 11 p.

We survey recent results related to the concentration of eigenfunctions. We also prove some new results concerning ball-concentration, as well as showing that eigenfunctions saturating lower bounds for L 1 -norms must also, in a measure theoretical sense, have extreme concentration near a geodesic.

DOI : https://doi.org/10.5802/jedp.638
Classification : 58J51,  35A99,  42B37
Mots clés : Eigenfunctions, Kakeya-Nikodym averages
@article{JEDP_2015____A9_0,
     author = {Sogge, Christopher D.},
     title = {Problems related to the concentration of eigenfunctions},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     eid = {9},
     publisher = {Groupement de recherche 2434 du CNRS},
     year = {2015},
     doi = {10.5802/jedp.638},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jedp.638/}
}
 Sogge, Christopher D. Problems related to the concentration of eigenfunctions. Journées équations aux dérivées partielles (2015), article  no. 9, 11 p. doi : 10.5802/jedp.638. http://www.numdam.org/articles/10.5802/jedp.638/

[1] Bérard, P. On the wave equation on a compact Riemannian manifold without conjugate points, Math. Z., Volume 155 (1977) no. 3, pp. 249-276 | MR 455055 | Zbl 0341.35052

[2] Blair, M. D.; Sogge, C. D. Concerning Topogonov’s Theorem and logarithmic improvement of estimates of eigenfunctions (preprint)

[3] Blair, M. D.; Sogge, C. D. Refined and microlocal Kakeya-Nikodym bounds of eigenfunctions in higher dimensions (preprint)

[4] Blair, M. D.; Sogge, C. D. On Kakeya–Nikodym averages, L p -norms and lower bounds for nodal sets of eigenfunctions in higher dimensions, J. Eur. Math. Soc. (JEMS), Volume 17 (2015) no. 10, pp. 2513-2543 | MR 3420515

[5] Blair, M. D.; Sogge, C. D. Refined and microlocal Kakeya-Nikodym bounds for eigenfunctions in two dimensions, Anal. PDE, Volume 8 (2015) no. 3, pp. 747-764 | Article | MR 3353830

[6] Bourgain, J. Geodesic restrictions and L p -estimates for eigenfunctions of Riemannian surfaces, Linear and complex analysis (Amer. Math. Soc. Transl. Ser. 2), Volume 226, Amer. Math. Soc., Providence, RI, 2009, pp. 27-35 | MR 2500507 | Zbl 1189.58015

[7] Burq, N.; Gérard, P.; Tzvetkov, N. Restrictions of the Laplace-Beltrami eigenfunctions to submanifolds, Duke Math. J., Volume 138 (2007) no. 3, pp. 445-486 | Article | MR 2322684 | Zbl 1131.35053

[8] Chen, X.; Sogge, C. D. A few endpoint geodesic restriction estimates for eigenfunctions, Comm. Math. Phys., Volume 329 (2014) no. 2, pp. 435-459 | Article | MR 3210140 | Zbl 1293.58011

[9] Colding, T. H.; Minicozzi, W. P. II Lower bounds for nodal sets of eigenfunctions, Comm. Math. Phys., Volume 306 (2011) no. 3, pp. 777-784 | Article | MR 2825508 | Zbl 1238.58020

[10] Colin de Verdière, Y. Ergodicité et fonctions propres du laplacien, Comm. Math. Phys., Volume 102 (1985) no. 3, pp. 497-502 http://projecteuclid.org/euclid.cmp/1104114465 | MR 818831 | Zbl 0592.58050

[11] Han, X. Small scale quantum ergodicity on negatively curved manifolds (2014) (http://arxiv.org/abs/1410.3911) | MR 3403398

[12] Hassell, A.; Tacy, M. Improvement of eigenfunction estimates on manifolds of nonpositive curvature, Forum Math., Volume 27 (2015) no. 3, pp. 1435-1451 | MR 3341481

[13] Hezari, H.; Rivière, G. L p norms, nodal sets, and quantum ergodicity (2014) (http://arxiv.org/abs/1411.4078)

[14] Hezari, H.; Sogge, C. D. A natural lower bound for the size of nodal sets, Anal. PDE, Volume 5 (2012) no. 5, pp. 1133-1137 | Article | MR 3022851

[15] Lester, S.; Rudnick, Z. Small scale equidistribution of eigenfunctions on the torus (http://arxiv.org/abs/1508.01074)

[16] Safarov, Y. G. Asymptotics of a spectral function of a positive elliptic operator without a nontrapping condition, Funktsional. Anal. i Prilozhen., Volume 22 (1988) no. 3, p. 53-65, 96 | MR 961761 | Zbl 0679.35074

[17] Šnirelʼman, A. I. Ergodic properties of eigenfunctions, Uspehi Mat. Nauk, Volume 29 (1974) no. 6(180), p. 181-182 | MR 402834 | Zbl 0324.58020

[18] Sogge, C. D. Localized L p -estimates of eigenfunctions: A note on an article of Hezari and Rivière (http://arxiv.org/abs/1503.07238) | MR 3439690

[19] Sogge, C. D. Oscillatory integrals and spherical harmonics, Duke Math. J., Volume 53 (1986) no. 1, pp. 43-65 | Article | MR 835795 | Zbl 0636.42018

[20] Sogge, C. D. Concerning the L p norm of spectral clusters for second-order elliptic operators on compact manifolds, J. Funct. Anal., Volume 77 (1988) no. 1, pp. 123-138 | Article | MR 930395 | Zbl 0641.46011

[21] Sogge, C. D. Fourier integrals in classical analysis, Cambridge Tracts in Mathematics, 105, Cambridge University Press, Cambridge, 1993, pp. x+237 | Article | MR 1205579 | Zbl 0783.35001

[22] Sogge, C. D. Kakeya-Nikodym averages and L p -norms of eigenfunctions, Tohoku Math. J. (2), Volume 63 (2011) no. 4, pp. 519-538 | Article | MR 2872954 | Zbl 1234.35156

[23] Sogge, C. D. Hangzhou lectures on eigenfunctions of the Laplacian, Annals of Mathematics Studies, 188, Princeton University Press, Princeton, NJ, 2014, pp. xii+193 | Article | MR 3186367

[24] Sogge, C. D.; Toth, J. A.; Zelditch, S. About the blowup of quasimodes on Riemannian manifolds, J. Geom. Anal., Volume 21 (2011) no. 1, pp. 150-173 | Article | MR 2755680 | Zbl 1214.58012

[25] Sogge, C. D.; Zelditch, S. Focal points and sup-norms of eigenfunctions (Rev. Mat. Iberoamericana, to appear.)

[26] Sogge, C. D.; Zelditch, S. Focal points and sup-norms of eigenfunctions manifolds II: the two-dimensional case (Rev. Mat. Iberoamericana, to appear.)

[27] Sogge, C. D.; Zelditch, S. Riemannian manifolds with maximal eigenfunction growth, Duke Math. J., Volume 114 (2002) no. 3, pp. 387-437 | Article | MR 1924569 | Zbl 1018.58010

[28] Sogge, C. D.; Zelditch, S. Lower bounds on the Hausdorff measure of nodal sets, Math. Res. Lett., Volume 18 (2011) no. 1, pp. 25-37 | Article | MR 2770580 | Zbl 1242.58017

[29] Sogge, C. D.; Zelditch, S. Lower bounds on the Hausdorff measure of nodal sets II, Math. Res. Lett., Volume 19 (2012) no. 6, pp. 1361-1364 | Article | MR 3091613 | Zbl 1283.58020

[30] Sogge, C. D.; Zelditch, S. On eigenfunction restriction estimates and L 4 -bounds for compact surfaces with nonpositive curvature, Advances in analysis: the legacy of Elias M. Stein (Princeton Math. Ser.), Volume 50, Princeton Univ. Press, Princeton, NJ, 2014, pp. 447-461 | MR 3329861

[31] Zelditch, S. Uniform distribution of eigenfunctions on compact hyperbolic surfaces, Duke Math. J., Volume 55 (1987) no. 4, pp. 919-941 | Article | MR 916129 | Zbl 0643.58029

[32] Zelditch, S. On the rate of quantum ergodicity. I. Upper bounds, Comm. Math. Phys., Volume 160 (1994) no. 1, pp. 81-92 http://projecteuclid.org/euclid.cmp/1104269516 | MR 1262192 | Zbl 0788.58043