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

Abstract

Let $ψ$ be an $L^2$-normalized Hecke-Maass form with a large spectral parameter $λ>0$ on a compact arithmetic congruence hyperbolic 3-manifold $X=Γ\backslash\mathrm{SL}(2,\mathbb{C})/\mathrm{SU}(2)$, and let $Y$ be a totally geodesic surface in $X$ with bounded diameter. The local $L^2$-bound for the restriction of $ψ$ to $Y$ is $\|ψ|_Y\|_{L^2(Y)}\ll λ^{1/4}$ by Burq, Gérard, 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 $ψ|_Y$ against geodesic beams over $Y$ via two amplification arguments. Combining these estimates, we can improve the local bound for generalized Fourier coefficients of $ψ|_Y$ against eigenfunctions on $Y$ with spectral parameters near $λ$. We also apply the amplification method to obtain a power saving over the trivial bound $O(1)$ for $L^2$-norms of $ψ$ restricted to $λ^{-1/2}$-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 $ψ$ by Sogge for $2<p<4$ from our improved bound for the Kakeya-Nikodym norm.