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

TL;DR

This paper establishes bounds on the restriction norms of Laplace eigenfunctions on hyperbolic surfaces along geodesics, using automorphic representation theory, advancing understanding of eigenfunction behavior in geometric analysis.

Contribution

It introduces new bounds on eigenfunction restrictions on hyperbolic surfaces utilizing automorphic representation theory, a novel approach in this context.

Findings

Established non-trivial L^2-norm bounds for eigenfunction restrictions

Applied automorphic function theory to geometric analysis problems

Connected eigenfunction restrictions to representation theory of PGL(2,R)

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).