An improvement on eigenfunction restriction estimates for compact boundaryless Riemannian manifolds with nonpositive sectional curvature

TL;DR

This paper improves eigenfunction restriction estimates on compact boundaryless Riemannian manifolds with nonpositive sectional curvature, especially for geodesic restrictions in two dimensions, using universal covers and Hadamard parametrix techniques.

Contribution

It provides new, sharper $L^p$ restriction estimates for eigenfunctions on manifolds with nonpositive curvature, extending previous results to broader cases and dimensions.

Findings

Improved $L^p$ restriction estimates for eigenfunctions on nonpositively curved manifolds.

Sharp bounds for restrictions to geodesics in 2D cases.

Extension of estimates to higher dimensions with specific $p$ ranges.

Abstract

Let $(M,g)$ be an $n$-dimensional compact boudaryless Riemannian manifold with nonpositive sectional curvature, then our conclusion is that we can give improved estimates for the $L^p$ norms of the restrictions of eigenfunctions to smooth submanifolds of dimension $k$, for $p>\dfrac{2n}{n-1}$ when $k=n-1$ and $p>2$ when $k\leq n-2$, compared to the general results of Burq, G\'erard and Tzvetkov \cite{burq}. Earlier, B\'erard \cite{Berard} gave the same improvement for the case when $p=\infty$, for compact Riemannian manifolds without conjugate points for $n=2$, or with nonpositive sectional curvature for $n\geq3$ and $k=n-1$. In this paper, we give the improved estimates for $n=2$, the $L^p$ norms of the restrictions of eigenfunctions to geodesics. Our proof uses the fact that, the exponential map from any point in $x\in M$ is a universal covering map from $\mathbb{R}^2\backsimeq T_{x}M$ to $M$, which allows us to lift the calculations up to the universal cover $(\mathbb{R}^2,\tilde{g})$, where $\tilde{g}$ is the pullback of $g$ via the exponential map. Then we prove the main estimates by using the Hadamard parametrix for the wave equation on $(\mathbb{R}^2,\tilde{g})$, the stationary phase estimates, and the fact that the principal coefficient of the Hadamard parametrix is bounded, by observations of Sogge and Zelditch in \cite{SZ}. The improved estimates also work for $n\geq 3$, with $p>\frac{4k}{n-1}$. We can then get the full result by interpolation.