Norms of geodesic restrictions for eigenfunctions on hyperbolic surfaces and representation theory

Abstract

We consider restrictions along closed geodesics and geodesic circles for eigenfunctions of the Laplace-Beltrami operator on a compact hyperbolic Riemann surface. We obtain a non-trivial bound on the L^2-norm of such restrictions as the eigenvalue tends to infinity. We use methods from the theory of automorphic functions and in particular the uniqueness of invariant functionals on irreducible unitary representations of PGL(2,R).

An updated version of the text not intended for a publication for being obsolete. A remark on a bound for a period added.

References in corpus (2)

• Multilinear eigenfunction estimates for the Laplace spectral projectors on compact manifolds
• Estimates of automorphic functions

Cited by in corpus (16)

• The subconvexity problem for $\GL_{2}$
• Distribution of mass of holomorphic cusp forms
• Nodal domains of Maass forms I
• Concerning the $L^4$ norms of typical eigenfunctions on compact surfaces
• Quantum ergodic restriction theorems, II: manifolds without boundary
• Estimates for the restriction of automorphic forms on hyperbolic manifolds to compact geodesic cycles
• Improved critical eigenfunction restriction estimates on Riemannian manifolds with constant negative curvature
• Nodal domains of Maass forms II
• Equidistribution of Eisenstein series on geodesic segments
• On integrals of eigenfunctions over geodesics
• An improvement on eigenfunction restriction estimates for compact boundaryless Riemannian manifolds with nonpositive sectional curvature
• Matrix elements of Fourier Integral Operators
• Sharp endpoint estimates for eigenfunctions restricted to submanifolds of codimension 2
• Quantum ergodic restriction theorems, I: interior hypersurfaces in domains with ergodic billiards
• Semiclassical L^p Estimates of Quasimodes on Submanifolds
• Concentration of eigenfunctions near a concave boundary