An improvement on eigenfunction restriction estimates for compact boundaryless Riemannian manifolds with nonpositive sectional curvature

Abstract

Let ( M , g ) (M,g) be an n n -dimensional compact boundaryless Riemannian manifold with nonpositive sectional curvature. Then our conclusion is that we can give improved estimates for the L p L^p norms of the restrictions of eigenfunctions of the Laplace-Beltrami operator to smooth submanifolds of dimension k k , for p > 2 n n − 1 p>\dfrac {2n}{n-1} when k = n − 1 k=n-1 and p > 2 p>2 when k ≤ n − 2 k\leq n-2 , compared to the general results of Burq, Gérard and Tzvetkov. Earlier, Bérard gave the same improvement for the case when p = ∞ p=\infty , for compact Riemannian manifolds without conjugate points for n = 2 n=2 , or with nonpositive sectional curvature for n ≥ 3 n\geq 3 and k = n − 1 k=n-1 . In this paper, we give the improved estimates for n = 2 n=2 , the L p L^p norms of the restrictions of eigenfunctions to geodesics. Our proof uses the fact that the exponential map from any point in x ∈ M x\in M is a universal covering map from R 2 ⋍ T x M \mathbb {R}^2\backsimeq T_{x}M to M M , which allows us to lift the calculations up to the universal cover ( R 2 , g ~ ) (\mathbb {R}^2,\tilde {g}) , where g ~ \tilde {g} is the pullback of g g via the exponential map. Then we prove the main estimates by using the Hadamard parametrix for the wave equation on ( R 2 , g ~ ) (\mathbb {R}^2,\tilde {g}) , 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 n\geq 3 , with p > 4 k n − 1 p>\frac {4k}{n-1} . We can then get the full result by interpolation.