Restrictions of Maass forms on $\mathrm{SL}(2,\mathbb{C})$ to hyperbolic surfaces and geodesic tubes

Let $\psi$ be an $L^2$-normalized Hecke-Maass form with a large spectral parameter $\lambda>0$ on a compact arithmetic congruence hyperbolic 3-manifold $X=\Gamma\backslash\mathrm{SL}(2,\mathbb{C})/\mathrm{SU}(2)$, and let $Y$ be a totally geodesic surface in $X$ with bounded diameter. The local $L^2$-bound for the restriction of $\psi$ to $Y$ is $\|\psi|_Y\|_{L^2(Y)}\ll \lambda^{1/4}$ by Burq, G\'erard, and Tzvetkov. We apply the method of arithmetic amplification developed by Iwaniec and Sarnak to obtain a power saving over the local bound. The new feature in the proof is that we establish two different estimates for the integrals of $\psi|_Y$ against geodesic beams over $Y$ via two amplification arguments. Combining these estimates, we can improve the local bound for generalized Fourier coefficients of $\psi|_Y$ against eigenfunctions on $Y$ with spectral parameters near $\lambda$. We also apply the amplification method to obtain a power saving over the trivial bound $O(1)$ for $L^2$-norms of $\psi$ restricted to $\lambda^{-1/2}$-neighborhoods of unit-length geodesic segments. Consequently, by applying a result of Blair and Sogge, we obtain power savings over the local $L^p$-bounds of $\psi$ by Sogge for $2<p<4$ from our improved bound for the Kakeya-Nikodym norm.