Quantum ergodicity and $L^p$ norms of restrictions of eigenfunctions

TL;DR

This paper establishes new $L^p$ restriction estimates for eigenfunctions on manifolds, demonstrating improvements under quantum ergodicity assumptions, with implications for nodal domains and boundary eigenfunction behavior.

Contribution

It extends $L^p$ restriction estimates to quantum ergodic eigenfunctions, showing logarithmic and $o(1)$ improvements, and applies these results to eigenfunctions on manifolds with boundary and ergodic billiards.

Findings

Logarithmic improvements on negatively curved manifolds

$o(1)$ improvements for ergodic geodesic flows

Number of nodal domains tends to infinity in 2D ergodic billiards

Abstract

We prove an analogue of Sogge's local $L^p$ estimates for $L^p$ norms of restrictions of eigenfunctions to submanifolds, and use it to show that for quantum ergodic eigenfunctions one can get improvements of the results of Burq-G\'erard-Tzvetkov, Hu, and Chen-Sogge. The improvements are logarithmic on negatively curved manifolds (without boundary) and by $o(1)$ for manifolds (with or without boundary) with ergodic geodesic flows. In the case of ergodic billiards with piecewise smooth boundary, we get $o(1)$ improvements on $L^\infty$ estimates of Cauchy data away from a shrinking neighborhood of the corners, and as a result using the methods of Ghosh-Reznikov-Sarnak and Jung-Zelditch, we get that the number of nodal domains of two dimensional ergodic billiards tends to infinity as $\lambda \to \infty$. These results work only for a full density subsequence of any given orthonormal basis of eigenfunctions. We also present an extension of the $L^p$ estimates of Burq-G\'erard-Tzvetkov, Hu, and Chen-Sogge, for the restrictions of Dirichlet and Neumann eigenfunctions to compact submanifolds of the interior of manifolds with piecewise smooth boundary. This part does not assume ergodicity on the manifolds.