We use the Gauss-Bonnet theorem and the triangle comparison theorems of Rauch and Toponogov to show that on compact Riemannian surfaces of negative curvature period integrals of eigenfunctions e λ over geodesics go to zero at the rate of O((log λ) −1/2 ) if λ are their frequencies. As discussed in [4], no such result is possible in the constant curvature case if the curvature is ≥ 0. Notwithstanding, we also show that these bounds for period integrals are valid provided that integrals of the curvature over all geodesic balls of radius r ≤ 1 are pinched from above by −δr N for some fixed N and δ > 0. This allows, for instance, the curvature to be nonpositive and to vanish of finite order at a finite number of isolated points. Naturally, the above results also hold for the appropriate type of quasi-modes.

This article concerns joint asymptotics of Fourier coefficients of restrictions of Laplace eigenfunctions ϕ j of a compact Riemannian manifold to a submanifold H ⊂ M . We fix a number c ∈ (0, 1) and study the asymptotics of the thin sums,λ) := j,λj ≤λ k:|µ k −cλj |<ǫ H ϕ j ψ k dV H 2 where {λ j } are the eigenvalues of √ −∆ M , and {(µ k , ψ k )} are the eigenvalues, resp. eigenfunctions, of √ −∆ H . The inner sums represent the 'jumps' of N c ǫ,H (λ) and reflect the geometry of geodesic c-bi-angles with one leg on H and a second leg on M with the same endpoints and compatible initial tangent vectors ξ ∈ S c H M, π H ξ ∈ B * H, where π H ξ is the orthogonal projection of ξ to H. A c-bi-angle occurs when |πH ξ| |ξ| = c. Smoothed sums in µ k are also studied, and give sharp estimates on the jumps. The jumps themselves may jump as ǫ varies, at certain values of ǫ related to periodicities in the c-bi-angle geometry. Subspheres of spheres and certain subtori of tori illustrate these jumps. The results refine those of the previous article [WXZ20] where the inner sums run over k : | µ k λj − c| ≤ ǫ and where geodesic bi-angles do not play a role.

Abstract. We show that one can obtain improved L 4 geodesic restriction estimates for eigenfunctions on compact Riemannian surfaces with nonpositive curvature. We achieve this by adapting Sogge's strategy in [12]. We first combine the improved L 2 restriction estimate of Blair and Sogge [3] and the classical improved L ∞ estimate of Bérard to obtain an improved weak-type L 4 restriction estimate. We then upgrade this weak estimate to a strong one by using the improved Lorentz space estimate of Bak and Seeger [1]. This estimate improves the L 4 restriction estimate of Burq, Gérard and Tzvetkov [5] and Hu [8] by a power of (log log λ) −1 . Moreover, in the case of compact hyperbolic surfaces, we obtain further improvements in terms of (log λ) −1 by applying the ideas from [7] and [3]. We are able to compute various constants that appeared in [7] explicitly, by proving detailed oscillatory integral estimates and lifting calculations to the universal cover H 2 .

We expand the class of curves (ϕ 1 (t), ϕ 2 (t)), t ∈ [0, 1] for which the ℓ 2 decoupling conjecture holds for 2 ≤ p ≤ 6. Our class of curves includes all real-analytic regular curves with isolated points of vanishing curvature and all curves of the form (t, t 1+ν ) for ν ∈ (0, ∞).

In this paper, we establish the sharp k-broad estimate for a class of phase functions satisfying the homogeneous convex conditions. As an application, we obtain improved local smoothing estimates for the half-wave operator in dimensions n ≥ 3. As a byproduct, we also generalize the restriction estimates of Ou-Wang [22] to a broader class of phase functions.