Riemannian manifolds with maximal eigenfunction growth

Sogge, Christopher D.

Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2000-2001), Exposé no. 24, 16 p.

[A] V.I. Arnold, Mathematical methods of classical mechanics. Graduate Texts in 1998. | Zbl

[Av] V. I. Avakumovič, Über die Eigenfunktionen auf geshchlossenen Riemannschen Mannigfaltigkeiten. Math. Z. 65 (1956), 327–344. | MR | Zbl

[BB] L. Bérard-Bergery, Quelques exemples de variétés riemanniennes où toutes les géodésiques issues d’un point sont fermés et de même longueur, suivis de quelques résultats sur leur topologie. Ann. Inst. Fourier (Grenoble) 27 (1977), 231–249. | Numdam | Zbl

[Besse] A. L. Besse, Manifolds all of whose geodesics are closed. With appendices by D. B. A. Epstein, J.-P. Bourguignon, L. Bérard-Bergery, M. Berger and J. L. Kazdan. Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], 93. Springer-Verlag, Berlin-New York, 1978. | MR | Zbl

[BM] E. Bierstone and P.D. Milman, Semianalytic and subanalytic sets. Inst. Hautes Études Sci. Publ. Math. No. 67 (1988), 5–42. | Numdam | MR | Zbl

[BC] A. Bonami and J.-C. Clerc, Sommes de Cesro et multiplicateurs des développements en harmoniques sphériques, Trans. Amer. Math. Soc. 183 (1973), 223–263. | MR | Zbl

[B] J. Bourgain, Eigenfunction bounds for compact manifolds with integrable geodesic flow, IHES preprint.

[CS] L. Carleson and P. Sjölin, Oscillatory integrals and a multiplier problem for the disc, Studia Math. 44 (1972), 287–299. | MR | Zbl

[CV] Y. Colin de Verdière, Spectre conjoint d’opérateurs pseudo-différentiels qui commutent. II. Le cas intégrable, Math. Z. 171 (1980), no. 1, 51–73. | Zbl

[CVV] Y. Colin de Verdiere and S. Vu Ngoc, Singular Bohr-Sommerfeld rules for 2D integrable systems, http://www-fourier.ujf-grenoble.fr/prepublications.html

[DG] J. J. Duistermaat and V. W. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics. Invent. Math. 29 (1975), 39–79. | MR | Zbl

[H] R. M. Hardt, Stratification via corank one projections. Singularities, Part 1 (Arcata, Calif., 1981), 559–566, Proc. Sympos. Pure Math., 40, Amer. Math. Soc., Providence, R.I., 1983. | MR | Zbl

[Ho 1] L. Hörmander, The spectral function of an elliptic operator. Acta Math. 121 (1968), 193–218. | MR | Zbl

[Ho 2] L. Hörmander, Oscillatory integrals and multipliers on $F{L}^p$. Ark. Mat. 11, 1–11. (1973). | MR | Zbl

[Ho IV] L. Hörmander, The Analysis of Linear Partial Differential Operators, Volume IV, Springer-Verlag Berlin Heidelberg, 1983. | Zbl

[Iv1] V. Ivriǐ, The second term of the spectral asymptotics for a Laplace Beltrami operator on manifolds with boundary. (Russian) Funktsional. Anal. i Prilozhen. 14 (1980), no. 2, 25–34. | MR | Zbl

[J] D. Jerison, Carleman inequalities for the Dirac and Laplace operators and unique continuation, Adv. in Math. 62 (1986), no. 2, 118–134. | MR | Zbl

[JK] D. Jerison and C. Kenig, Unique continuation and absence of positive eigenvalues for Schrödinger operators, Ann. of Math. (2) 121 (1985), no. 3, 463–494. | MR | Zbl

[K] W. Klingenberg, Riemannian Geometry, de Gruyter Studies in Mathematics 1, de Gruyter, New York (1982). | MR | Zbl

[KMS] D. Kosygin, A. Minasov, Ya. G. Sinaǐ, Statistical properties of the spectra of Laplace-Beltrami operators on Liouville surfaces. (Russian) Uspekhi Mat. Nauk 48 (1993), no. 4(292), 3–130; translation in Russian Math. Surveys 48 (1993), no. 4, 1–142 | MR | Zbl

[Le] B. M. Levitan, On the asymptoptic behavior of the spectral function of a self-adjoint differential equaiton of second order. Isv. Akad. Nauk SSSR Ser. Mat. 16 (1952), 325–352. | MR | Zbl

[L] S. Lojasiewicz, “Ensembles semi-analytiques”, Preprint, I.H.E.S., Bures-sur-Yvette, 1964.

[MS] W. Minicozzi and C. D. Sogge, Negative results for Nikodym maximal functions and related oscillatory integrals in curved space. Math. Res. Lett. 4 (1997), no. 2-3, 221–237. | MR | Zbl

[Saf] Y. Safarov, Asymptotics of a spectral function of a positive elliptic operator without a nontrapping condition. (Russian) Funktsional. Anal. i Prilozhen. 22 (1988), no. 3,53–65, 96 translation in Funct. Anal. Appl. 22 (1988), no. 3, 213–223 (1989). | MR | Zbl

[S] L. Simon, Lecture notes (unpublished, 1992).

[So1] C. D. Sogge, Oscillatory integrals and spherical harmonics. Duke Math. J. 53 (1986), 43–65. | MR | Zbl

[So2] C. D. Sogge, Concerning the $L^p$ norm of spectral clusters for second order elliptic operators on compact manifolds. J. Funct. Anal. 77 (1988), 123–134. | MR | Zbl

[So3] C. D. Sogge, On the convergence of Riesz means on compact manifolds. Ann. of Math. (2) 126 (1987), no. 2, 439–447. | MR | Zbl

[So4] C. D. Sogge, Fourier integrals in classical analysis, Cambridge Univ. Press, Cambridge, 1993. | MR | Zbl

[SZ] C. D. Sogge, and S. Zelditch, Riemannian manifolds with maximal eigenfunction growth, http://front.math.ucdavis.edu/math.AP/0103172 .

[St] E. M. Stein, Oscillatory integrals in Fourier analysis, Beijing lectures in harmonic analysis (Beijing, 1984), 307–355, Ann. of Math. Stud., 112, Princeton Univ. Press, Princeton, NJ, 1986. | MR | Zbl

[TZ] J.A.Toth and S.Zelditch, Sup norms of eigenfunctions in the completely integrable case (preprint, 2000).

[V] J. VanderKam, $L^{\infty }$ norms and quantum ergodicity on the sphere. Internat. Math. Res. Notices (1997), 329–347; Correction to: "$L^{\infty }$ norms and quantum ergodicity on the sphere", Internat. Math. Res. Notices (1998), 65. | MR | Zbl

[Z] S. Zelditch, The inverse spectral problem for surfaces of revolution, JDG 49 (1998), 207-264. | MR | Zbl