On logarithmic improvements of critical geodesic restriction bounds in the presence of nonpositive curvature
- 1 Downloads
We consider upper bounds on the growth of L p norms of restrictions of eigenfunctions and spectral clusters to geodesic segments in a nonpositively curved manifold in the high frequency limit. This sharpens results of Chen and Sogge as well as Xi and Zhang, which showed that the crux of the problem is to establish bounds on the mixed partials of the distance function on the covering manifold restricted to geodesic segments. The innovation in this work is the development of a formula for the third variation of arc length on the covering manifold, which allows for a coordinate free expressions of these mixed partials.
Unable to display preview. Download preview PDF.
- [BS15a]M. D. Blair and C. D. Sogge, Concerning Topogonov’s Theorem and logarithmic improvement of estimates of eigenfunctions, Journal of Differential Geometry, to appear, arXiv:1510.07726.Google Scholar
- [Hez16]H. Hezari, Quantum ergodicity and Lp norms of restrictions of eigenfunctions, preprint, arXiv:1606.08066 (2016).Google Scholar
- [Lee97]J. M. Lee, Riemannian Manifolds, Graduate Texts in Mathematics, Vol. 176, Springer-Verlag, New York, 1997.Google Scholar
- [Rez04]A. Reznikov, Norms of geodesic restrictions for eigenfunctions on hyperbolic surfaces and representation theory, preprint, arXiv math/0403437 (2004).Google Scholar
- [Sog14]C. D. Sogge, Hangzhou Lectures on Eigenfunctions of the Laplacian, Annals of Mathematics Studies, Vol. 188, Princeton University Press, Princeton, NJ, 2014.Google Scholar
- [Ste93]E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, Vol. 43, Princeton University Press, Princeton, NJ, 1993.Google Scholar
- [SZ14]C. D. Sogge and S. Zelditch, On eigenfunction restriction estimates and L4-bounds for compact surfaces with nonpositive curvature, in Advances in Analysis: The Legacy of Elias M. Stein, Princeton Mathematical Series, Vol. 50, Princeton University Press, Princeton, NJ, 2014, pp. 447–461.MATHGoogle Scholar
- [Tac16]M. Tacy, A note on constructing sharp examples for Lp norms of eigenfunctions and quasimodes near submanifolds, preprint, arXiv:1605.03698 (2016).Google Scholar