Eigenfunction restriction estimates for curves with nonvanishing geodesic curvatures in compact Riemannian surfaces with nonpositive curvature
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- by Chamsol Park;
- Trans. Amer. Math. Soc. 376 (2023), 5809-5855
- DOI: https://doi.org/10.1090/tran/8948
- Published electronically: May 23, 2023
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Abstract:
For $2\leq p<4$, we study the $L^p$ norms of restrictions of eigenfunctions of the Laplace-Beltrami operator on smooth compact $2$-dimensional Riemannian manifolds. Burq, Gérard, and Tzvetkov [Duke Math. J. 138 (2007), pp. 445–486], and Hu [Forum Math. 21 (2009), pp. 1021–1052] found the eigenfunction estimates restricted to a curve with nonvanishing geodesic curvatures. We will explain how the proof of the known estimates helps us to consider the case where the given smooth compact Riemannian manifold has nonpositive sectional curvatures. For $p=4$, we will also obtain a logarithmic analogous estimate, by using arguments in Xi and Zhang [Comm. Math. Phys. 350 (2017), pp. 1299–1325], Sogge [Math. Res. Lett. 24 (2017), pp. 549–570], and Bourgain [Geom. Funct. Anal. 1 (1991), pp. 147–187].References
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Bibliographic Information
- Chamsol Park
- Affiliation: Department of Mathematics and Statistics, University of New Mexico, Albuquerque, New Mexico 87131
- ORCID: 0009-0004-5917-1282
- Email: parkcs@unm.edu
- Received by editor(s): March 15, 2022
- Received by editor(s) in revised form: February 4, 2023
- Published electronically: May 23, 2023
- Additional Notes: The author was supported in part by the National Science Foundation grants DMS-1301717 and DMS-1565436.
- © Copyright 2023 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 376 (2023), 5809-5855
- MSC (2020): Primary 58J40; Secondary 35S30, 42B37
- DOI: https://doi.org/10.1090/tran/8948
- MathSciNet review: 4630760