Eigenfunction restriction estimates for curves with nonvanishing geodesic curvatures in compact Riemannian surfaces with nonpositive curvature

Park, Chamsol

doi:10.1090/tran/8948

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].

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