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

We show that one can obtain logarithmic improvements of L2 geodesic restriction estimates for eigenfunctions on 3-dimensional compact Riemannian manifolds with constant negative curvature. We obtain a (log⁡λ)−12 gain for the L2-restriction bounds, which improves the corresponding bounds of Burq, Gérard and Tzvetkov [4], Hu [10], Chen and Sogge [6]. We achieve this by adapting the approaches developed by Chen and Sogge [6], Blair and Sogge [3], Xi and the author [19]. 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 [6]. We prove detailed oscillatory integral estimates with fold singularities by Phong and Stein [12] and use the Poincaré half-space model to establish bounds for various derivatives of the distance function restricted to geodesic segments on the universal cover H3.