On eigenfunction restriction estimates and $L^4$-bounds for compact surfaces with nonpositive curvature

TL;DR

This paper improves eigenfunction restriction estimates and $L^4$ bounds on compact surfaces with nonpositive curvature by leveraging universal covers and the Hadamard parametrix, leading to sharper $L^2$ restriction bounds.

Contribution

It introduces new restriction estimates for eigenfunctions on nonpositively curved surfaces using universal cover techniques and the Hadamard parametrix, enhancing previous bounds.

Findings

Improved $L^2$ restriction estimates for eigenfunctions on nonpositive curvature surfaces.

Utilization of universal cover and Hadamard parametrix for sharper bounds.

Application of G"unther's comparison theorem to control volume elements.

Abstract

Let $(M,g)$ be a two-dimensional compact boundaryless Riemannian manifold with nonpostive curvature, then we shall give improved estimates for the $L^2$-norms of the restrictions of eigenfunctions to unit-length geodesics, compared to the general results of Burq, G\'erard and Tzvetkov \cite{burq}. By earlier results of Bourgain \cite{bourgainef} and the first author \cite{Sokakeya}, they are equivalent to improvements of the general $L^p$-estimates in \cite{soggeest} for $n=2$ and $2<p<6$. The proof uses the fact that the exponential map from any point in $x_0\in M$ is a universal covering map from $\Rt \simeq T_{x_0}M$ to $M$ (the Cartan-Hadamard- von Mangolt theorem), which allows us to lift the necessary calculations up to the universal cover $(\Rt, \tilde g)$ where $\tilde g$ is the pullback of $g$ via the exponential map. We then prove the main estimates by using the Hadamard parametrix for the wave equation on $(\Rt, \tilde g)$ and the fact that the classical comparison theorem of G\"unther \cite{Gu} for the volume element in spaces of nonpositive curvature gives us desirable bounds for the principal coefficient of the Hadamard parametrix, allowing us to prove our main result.