Improved critical eigenfunction restriction estimates on Riemannian manifolds with constant negative curvature

TL;DR

This paper demonstrates logarithmic improvements in $L^2$ geodesic restriction estimates for eigenfunctions on 3D negatively curved manifolds, utilizing explicit wave kernel formulas and advanced oscillatory integral estimates.

Contribution

It introduces a new explicit wave kernel formula for 3D hyperbolic space and applies refined oscillatory integral estimates to improve eigenfunction restriction bounds.

Findings

Achieves a $( ext{log}\lambda)^{-rac{1}{2}}$ gain in restriction estimates.

Provides explicit formulas for wave kernels on hyperbolic space.

Improves previous bounds by adapting advanced oscillatory integral techniques.

Abstract

We show that one can obtain logarithmic improvements of $L^2$ geodesic restriction estimates for eigenfunctions on 3-dimensional compact Riemannian manifolds with constant negative curvature. We obtain a $(\log\lambda)^{-\frac12}$ gain for the $L^2$-restriction bounds, which improves the corresponding bounds of Burq, G\'erard and Tzvetkov, Hu, Chen and Sogge. We achieve this by adapting the approaches developed by Chen and Sogge, Blair and Sogge, Xi and the author. We derive an explicit formula for the wave kernel on 3D hyperbolic space, which improves the kernel estimates from the Hadamard parametrix in Chen and Sogge. We prove detailed oscillatory integral estimates with fold singularities by Phong and Stein and use the Poincar\'e half-space model to establish bounds for various derivatives of the distance function restricted to geodesic segments on the universal cover $\mathbb{H}^3$.