AN IMPROVEMENT ON EIGENFUNCTION RESTRICTION ESTIMATES FOR COMPACT BOUNDARYLESS RIEMANNIAN MANIFOLDS WITH NONPOSITIVE SECTIONAL CURVATURE

Let (M, g) be an n-dimensional compact boundaryless Riemannian manifold with nonpositive sectional curvature. Then our conclusion is that we can give improved estimates for the L-p norms of the restrictions of eigenfunctions of the Laplace-Beltrami operator to smooth submanifolds of dimension k, for p > 2n/n -1 when k = n-1 and p > 2 when k <= n-2, compared to the general results of Burq, Gerard and Tzvetkov. Earlier, Berard gave the same improvement for the case when p = infinity, for compact Riemannian manifolds without conjugate points for n = 2, or with nonpositive sectional curvature for n >= 3 and k = n -1. In this paper, we give the improved estimates for n = 2, the L-p norms of the restrictions of eigenfunctions to geodesics. Our proof uses the fact that the exponential map from any point in x is an element of M is a universal covering map from R-2 similar or equal to TxM to M, which allows us to lift the calculations up to the universal cover (R-2, (g) over tilde), where (g) over tilde is the pullback of g via the exponential map. Then we prove the main estimates by using the Hadamard parametrix for the wave equation on (R-2, (g) over tilde), the stationary phase estimates, and the fact that the principal coefficient of the Hadamard parametrix is bounded, by observations of Sogge and Zelditch. The improved estimates also work for n >= 3, with p > 4k/n-1. We can then get the full result by interpolation.