An improvement on eigenfunction restriction estimates for compact boundaryless Riemannian manifolds with nonpositive sectional curvature
HTML articles powered by AMS MathViewer
- by Xuehua Chen PDF
- Trans. Amer. Math. Soc. 367 (2015), 4019-4039 Request permission
Abstract:
Let $(M,g)$ be an $n$-dimensional compact boundaryless Riemannian manifold with nonpositive sectional curvature. Then our conclusion is that we can give improved estimates for the $L^p$ norms of the restrictions of eigenfunctions of the Laplace-Beltrami operator to smooth submanifolds of dimension $k$, for $p>\dfrac {2n}{n-1}$ when $k=n-1$ and $p>2$ when $k\leq n-2$, compared to the general results of Burq, Gérard and Tzvetkov. Earlier, Bérard gave the same improvement for the case when $p=\infty$, for compact Riemannian manifolds without conjugate points for $n=2$, or with nonpositive sectional curvature for $n\geq 3$ and $k=n-1$. In this paper, we give the improved estimates for $n=2$, the $L^p$ norms of the restrictions of eigenfunctions to geodesics. Our proof uses the fact that the exponential map from any point in $x\in M$ is a universal covering map from $\mathbb {R}^2\backsimeq T_{x}M$ to $M$, which allows us to lift the calculations up to the universal cover $(\mathbb {R}^2,\tilde {g})$, where $\tilde {g}$ is the pullback of $g$ via the exponential map. Then we prove the main estimates by using the Hadamard parametrix for the wave equation on $(\mathbb {R}^2,\tilde {g})$, the stationary phase estimates, and the fact that the principal coefficient of the Hadamard parametrix is bounded, by observations of Sogge and Zelditch. The improved estimates also work for $n\geq 3$, with $p>\frac {4k}{n-1}$. We can then get the full result by interpolation.References
- Pierre H. Bérard, On the wave equation on a compact Riemannian manifold without conjugate points, Math. Z. 155 (1977), no. 3, 249–276. MR 455055, DOI 10.1007/BF02028444
- N. Burq, P. Gérard, and N. Tzvetkov, Restrictions of the Laplace-Beltrami eigenfunctions to submanifolds, Duke Math. J. 138 (2007), no. 3, 445–486 (English, with English and French summaries). MR 2322684, DOI 10.1215/S0012-7094-07-13834-1
- Paul Günther, Einige Sätze über das Volumenelement eines Riemannschen Raumes, Publ. Math. Debrecen 7 (1960), 78–93 (German). MR 141058
- Jacques Hadamard, Lectures on Cauchy’s problem in linear partial differential equations, Dover Publications, New York, 1953. MR 0051411
- A. Hassell and M. Tacey, personal communication.
- Lars Hörmander, The analysis of linear partial differential operators. I, Classics in Mathematics, Springer-Verlag, Berlin, 2003. Distribution theory and Fourier analysis; Reprint of the second (1990) edition [Springer, Berlin; MR1065993 (91m:35001a)]. MR 1996773, DOI 10.1007/978-3-642-61497-2
- Rui Hu, $L^p$ norm estimates of eigenfunctions restricted to submanifolds, Forum Math. 21 (2009), no. 6, 1021–1052. MR 2574146, DOI 10.1515/FORUM.2009.051
- Isaac Chavel, Riemannian geometry, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 98, Cambridge University Press, Cambridge, 2006. A modern introduction. MR 2229062, DOI 10.1017/CBO9780511616822
- A. Reznikov, Norms of geodesic restrictions for eigenfunctions on hyperbolic surfaces and representation theory, arXiv:math.AP/0403437.
- 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
- Christopher D. Sogge, Fourier integrals in classical analysis, Cambridge Tracts in Mathematics, vol. 105, Cambridge University Press, Cambridge, 1993. MR 1205579, DOI 10.1017/CBO9780511530029
- C. D. Sogge, Hangzhou lectures on eigenfunctions of the Laplacian (in preparation), www.mathematics.jhu.edu/sogge/zju.
- Christopher D. Sogge, Kakeya-Nikodym averages and $L^p$-norms of eigenfunctions, Tohoku Math. J. (2) 63 (2011), no. 4, 519–538. MR 2872954, DOI 10.2748/tmj/1325886279
- Christopher D. Sogge, John A. Toth, and Steve Zelditch, About the blowup of quasimodes on Riemannian manifolds, J. Geom. Anal. 21 (2011), no. 1, 150–173. MR 2755680, DOI 10.1007/s12220-010-9168-6
- Christopher D. Sogge and Steve Zelditch, Riemannian manifolds with maximal eigenfunction growth, Duke Math. J. 114 (2002), no. 3, 387–437. MR 1924569, DOI 10.1215/S0012-7094-02-11431-8
- C. Sogge and S. Zelditch, On eigenfunction restriction estimates and $L^4$-bounds for compact surfaces with nonpositive curvature.
- Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
Additional Information
- Xuehua Chen
- Affiliation: Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218
- Email: xchen@math.jhu.edu
- Received by editor(s): May 12, 2012
- Received by editor(s) in revised form: November 9, 2012, and February 19, 2013
- Published electronically: September 18, 2014
- Additional Notes: The author would like to cordially thank her advisor, Christopher Sogge, for his generous help and unlimited patience.
- © Copyright 2014 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 367 (2015), 4019-4039
- MSC (2010): Primary 35F99; Secondary 35L20, 42C99
- DOI: https://doi.org/10.1090/S0002-9947-2014-06158-8
- MathSciNet review: 3324918