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Restrictions of the Laplace-Beltrami eigenfunctions to submanifolds. (English) Zbl 1131.35053

Let \((M,g)\) be a compact smooth Riemannian manifold of the dimension \(d\) without boundary and \(\Delta\) the associated with the metric \(g\) Laplace-Beltrami operator. Let \(\{ \varphi_\lambda \}\), \(\lambda \geq 0\) be the eigenfunctions of \(\Delta\), i.e. \(-\Delta\varphi_\lambda=\lambda^2\varphi_\lambda\). The authors study the possible growth of the \(L^p\)-norm, \(2 \leq p \leq +\infty\) of the restrictions of \(\varphi_\lambda\) to submanifolds \(\Sigma\) of \(M\) in the simplest case when \(\Sigma\) is a smooth curve \(\gamma :[a,b] \to M\) parametrized by arc length. When the curve is a geodetic it is shown that on the sphere the obtained estimates are sharp, optimal for the sphere.

MSC:

35P20 Asymptotic distributions of eigenvalues in context of PDEs
35J15 Second-order elliptic equations
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
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[1] N. Anantharaman, Entropy and the localization of eigenfunctions , to appear in Ann. of Math. (2), preprint, 2004. · Zbl 1175.35036
[2] N. Burq, P. GéRard, and N. Tzvetkov. Multilinear estimates for the Laplace spectral projector on compact manifolds , C. R. Math. Acad. Sci. Paris 338 (2004), 359–364. · Zbl 1040.58011
[3] -, Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces , Invent. Math. 159 (2005), 187–223. · Zbl 1092.35099
[4] -, Multilinear eigenfunction estimates and global existence for the three dimensional nonlinear Schrödinger equations , Ann. Sci. École Norm. Sup. (4) 38 (2005), 255–301. · Zbl 1116.35109
[5] Y. Colin De VerdièRe, Ergodicité et fonctions propres du laplacien , Comm. Math. Phys. 102 (1985), 497–502. · Zbl 0592.58050
[6] S. Gallot, D. Hulin, and J. Lafontaine, Riemannian Geometry , 2nd ed., Universitext, Springer, Berlin, 1990.
[7] P. GéRard and é. Leichtnam, Ergodic properties of eigenfunctions for the Dirichlet problem , Duke Math. J. 71 (1993), 559–607. · Zbl 0788.35103
[8] A. Greenleaf and A. Seeger, Fourier integral operators with fold singularities , J. Reine Angew. Math. 455 (1994), 35–56. · Zbl 0799.42008
[9] B. Helffer, A. Martinez, and D. Robert, Ergodicité et limite semi-classique , Comm. Math. Phys. 109 (1987), 313–326. · Zbl 0624.58039
[10] L. HöRmander, The spectral function of an elliptic operator , Acta Math. 121 (1968), 193–218. · Zbl 0164.13201
[11] -, The Analysis of Linear Partial Differential Operators, I , Grundlehren Math. Wiss. 256 , Springer, Berlin, 1983.
[12] -, The Analysis of Linear Partial Differential Operators, IV , Grundlehren Math. Wiss. 275 , Springer, Berlin, 1985.
[13] E. Lindenstrauss, Invariant measures and arithmetic quantum unique ergodicity , Ann. of Math. (2) 163 (2006), 165–219. · Zbl 1104.22015
[14] R. B. Melrose, Equivalence of glancing hypersurfaces , Invent. Math. 37 (1976), 165–191. · Zbl 0354.53033
[15] A. Reznikov, Norms of geodesic restrictions for eigenfunctions on hyperbolic surfaces and representation theory , preprint,\arxivmath/0403437v2[math.AP] · Zbl 1008.53041
[16] P. Sarnak, “Arithmetic quantum chaos” in The Schur lectures (Tel Aviv, 1992) , Israel Math. Conf. Proc. 8 , Bar-Ilan Univ., Ramat Gan, Israel, 1995, 183–236. · Zbl 0831.58045
[17] A. I. Shnirelman [šNirel\(^\prime\)Man], Ergodic properties of eigenfunctions (in Russian), Uspekhi Mat. Nauk 29 , no. 6 (1974), 181–182. · Zbl 0285.16024
[18] C. D. Sogge, Concerning the \(L^p\) norm of spectral clusters for second-order elliptic operators on compact manifolds, J. Funct. Anal. 77 (1988), 123–138. · Zbl 0641.46011
[19] -, Fourier Integrals in Classical Analysis , Cambridge Tracts in Math., Cambridge Univ. Press, Cambridge, 1993. · Zbl 0783.35001
[20] C. D. Sogge and S. Zelditch, Riemannian manifolds with maximal eigenfunction growth , Duke Math. J. 114 (2002), 387–437. · Zbl 1018.58010
[21] E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals , with the assistance of Timothy S. Murphy, Princeton Math. Ser. 43 , Monogr. Harmon. Analysis 3 , Princeton Univ. Press, Princeton, 1993. · Zbl 0821.42001
[22] G. Szegö, Orthogonal Polynomials , 4th ed., Amer. Math. Soc. Colloq. Publ. 23 , Amer. Math. Soc., Providence, 1974.
[23] D. Tataru, On the regularity of boundary traces for the wave equation , Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 26 (1998), 185–206. · Zbl 0932.35136
[24] M. E. Taylor, “Diffraction effects in the scattering of waves” in Singularities in Boundary Value Problems (Maratea, Italy, 1980) , NATO Adv. Study Inst. Ser. C: Math. Phys. Sci. 65 , Reidel, Dordrecht, Netherlands, 1981, 271–316. · Zbl 0482.35070
[25] S. Zelditch, Uniform distribution of eigenfunctions on compact hyperbolic surfaces , Duke Math. J. 55 (1987), 919–941. · Zbl 0643.58029
[26] S. Zelditch and M. Zworski, Ergodicity of eigenfunctions for ergodic billiards , Comm. Math. Phys. 175 (1996), 673–682. · Zbl 0840.58048
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