Quantum ergodicity and $L^p$ norms of restrictions of eigenfunctions
  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. 
Hamid Hezari
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