Geodesic restrictions for the Casimir operator

TL;DR

This paper establishes new bounds on the restriction of eigenfunctions of the hyperbolic Casimir operator to specific hypersurfaces in the tangent sphere bundle of a compact hyperbolic Riemann surface, surpassing previous local bounds.

Contribution

It provides the first nontrivial bounds on eigenfunction restrictions for the hyperbolic Casimir operator, extending beyond classical local bounds.

Findings

Established nontrivial L^2 restriction bounds for eigenfunctions

Surpassed known local bounds of L. Hormander

Applied to tangent sphere bundles of hyperbolic surfaces

Abstract

We consider the hyperbolic Casimir operator C defined on the tangent sphere bundle SY of a compact hyperbolic Riemann surface Y. We prove a nontrivial bound on the L^2-norm of the restriction of eigenfunctions of C to certain natural hypersurfaces in SY. The result that we obtain goes beyond known (sharp) local bounds of L. Hormander.