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

TL;DR

This work analyzes L^2-restrictions of L^2-normalized Hecke–Maass forms on a compact arithmetic hyperbolic 3-manifold X to a totally geodesic surface Y, achieving a power-saving improvement over the local bound by λ^{1/4−1/1220+ε}. The authors combine Harish-Chandra and Helgason harmonic analysis with arithmetic amplification à la Iwaniec–Sarnak, introducing a two-pronged amplification scheme that uses a geodesic-beam decomposition and controlled Hecke returns to beat the trivial bounds. A novel aspect is the decomposition into geodesic beams and two amplification mechanisms—one for general beams and one for beam pairs under Hecke correspondences—yielding new bounds for generalized Fourier coefficients and Kakeya–Nikodym tubes. These results feed into improved L^p-restriction bounds for 2<p<4 via Blair–Sogge-type Kakeya–Nikodym control, marking a significant sharpening of restriction phenomena in the arithmetic-hyperbolic setting. The techniques rely on a precise spectral/decomposition framework, leveraging the pretrace formula and orbit-counting estimates to bridge analysis on hyperbolic spaces with arithmetic data.

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.

Theorems & Definitions (126)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 3.1
• Theorem 3.2: eguchi1979asymptoticeguchi1977fourier
• Proposition 3.3
• Corollary 3.4
• Lemma 3.5
• proof
• Proposition 3.6