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

By A. Reznikov. 2004

50 citations recorded by the source corpus.

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).

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/bea8fd6ed306362a5a769dc130038f0bd4cb9aaf

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Kakeya-Nikodym averages and $L^p$-norms of eigenfunctions (2009, background): “unit length geodesics then (1.5) sup 2 Z je j j2 ds.1=2 j ke j k2 L2(M) ; j= 1;2;3;:::; which is sharp in view of (1.3). Related results for hyperbolic surfaces were obtained earlier by Reznikov [20], who opened up the present line of investigation. The proof of (1.5) boils down to bounds for cert”
Distribution of mass of holomorphic cusp forms (2012, background): “Reznikov [R1] initiated a study of restricted L(2)-norms along various curves for Maaß forms with large eigenvalue.”
A Few Endpoint Geodesic Restriction Estimates for Eigenfunctions (2012, unknown): “prove certain estimates of the form (1.1) sup γ∈Π Z |eλ|p ds 1 p ≤ C λ(1 +λ)σ(n,p)ke λkL2(M), with ds denoting arc-length measure on γ. Burq, G´erard and Tzvetkov [7] and Hu [19] (see also Reznikov [25] for earlier work) showed that one can take Cλ to be independent of λ > 1 and (1.2) σ(2,p) = ( ”
The subconvexity problem for GL2 (2009, result): “Gerard, and Tzvetkov [11], generalizing a result of Reznikov [ 59 ], asserts the following general bound”
Lp norm estimates of eigenfunctions restricted to submanifolds (2009, background): “60/61 (1990), 413–419 [13] Reznikov A.: Norms of geodesic restrictions on hyperbolic surfaces and representation theory. arXiv: math.”
On eigenfunction restriction estimates and $L^4$-bounds for compact surfaces with nonpositive curvature (2011, background): “ any 2-dimensional compact boundaryless Riemannian manifold one has (1.2) Z γ |eλ|2 ds 1/2 ≤ Cλ14ke λkL2(M), with Cindependent of γ∈ Π. The ﬁrst such estimates were somewhat weaker ones of Reznikov [13] for hyperbolic surfaces, which inspired this current line of research. The estimate (1.2) is shar”