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

TL;DR

This paper extends eigenfunction restriction estimates to curves with nonvanishing geodesic curvature on compact Riemannian surfaces with nonpositive curvature, including new logarithmic bounds at the critical exponent.

Contribution

It generalizes previous restriction estimates to nonpositive curvature surfaces and derives a logarithmic estimate at the critical exponent p=4.

Findings

Established restriction estimates for eigenfunctions on nonpositively curved surfaces.

Derived a logarithmic bound for p=4 using advanced harmonic analysis techniques.

Extended known results to a broader class of Riemannian manifolds.

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, G\'erard, and Tzvetkov \cite{BurqGerardTzvetkov2007restrictions}, and Hu \cite{Hu2009lp} 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 \cite{XiZhang2017improved}, Sogge \cite{Sogge2017ImprovedCritical}, and Bourgain \cite{Bourgain1991Besicovitch}.