A few endpoint geodesic restriction estimates for eigenfunctions

TL;DR

This paper establishes new endpoint geodesic restriction estimates for eigenfunctions on compact manifolds, improving bounds in 2D and 3D cases using harmonic analysis and geometric techniques.

Contribution

It introduces novel endpoint restriction bounds for eigenfunctions on compact manifolds, leveraging harmonic analysis and geometric properties, with improvements in specific curvature settings.

Findings

Improved $L^2$-restriction bounds in 3D manifolds.

Enhanced $L^4$-estimates for geodesic restrictions in 2D nonpositive curvature.

Further improvements in 3D with constant nonpositive curvature exploiting totally geodesic submanifolds.

Abstract

We prove a couple of new endpoint geodesic restriction estimates for eigenfunctions. In the case of general 3-dimensional compact manifolds, after a $TT^*$ argument, simply by using the $L^2$-boundedness of the Hilbert transform on $\R$, we are able to improve the corresponding $L^2$-restriction bounds of Burq, G\'erard and Tzvetkov and Hu. Also, in the case of 2-dimensional compact manifolds with nonpositive curvature, we obtain improved $L^4$-estimates for restrictions to geodesics, which, by H\"older's inequality and interpolation, implies improved $L^p$-bounds for all exponents $p\ge 2$. We do this by using oscillatory integral theorems of H\"ormander, Greenleaf and Seeger, and Phong and Stein, along with a simple geometric lemma (Lemma \ref{lemma3.2}) about properties of the mixed-Hessian of the Riemannian distance function restricted to pairs of geodesics in Riemannian surfaces. We are also able to get further improvements beyond our new results in three dimensions under the assumption of constant nonpositive curvature by exploiting the fact that in this case there are many totally geodesic submanifolds.