Abstract. Let (M, g) be a compact smooth Riemannian manifold. We obtain new off-diagonal estimates as λ → ∞ for the remainder in the pointwise Weyl Law for the kernel of the spectral projector of the Laplacian onto functions with frequency at most λ. A corollary is that, when rescaled around a non self-focal point, the kernel of the spectral projector onto the frequency interval (λ, λ + 1] has a universal scaling limit as λ → ∞ (depending only on the dimension of M ). Our results also imply that if M has no conjuage points, then immersions of M into Euclidean space by an orthonormal basis of eigenfunctions with frequencies in (λ, λ + 1] are embeddings for all λ sufficiently large.

This paper is dedicated to the study of the topologies and nesting configurations of the components of the zero set of monochromatic random waves. We prove that the probability of observing any diffeomorphism type and any nesting arrangement among the zero set components is strictly positive for waves of large enough frequencies. Our results are a consequence of building Laplace eigenfunctions in euclidean space whose zero sets have a component with prescribed topological type or an arrangement of components with prescribed nesting configuration. © 2018 Wiley Periodicals, Inc.

No abstract

Abstract. In this paper, we study conformal invariants that arise from nodal sets and negative eigenvalues of conformally covariant operators; more specifically, the GJMS operators, which include the Yamabe and Paneitz operators. We give several applications to curvature prescription problems. We establish a version in conformal geometry of Courant's Nodal Domain Theorem. We also show that on any manifold of dimension n ≥ 3, there exist many metrics for which our invariants are nontrivial. We prove that the Yamabe operator can have an arbitrarily large number of negative eigenvalues on any manifold of dimension n ≥ 3. We obtain similar results for some higher order GJMS operators on some Einstein and Heisenberg manifolds. We describe the invariants arising from the Yamabe and Paneitz operators associated to left-invariant metrics on Heisenberg manifolds. Finally, in the appendix, the 2nd named author and Andrea Malchiodi study the Q-curvature prescription problems for non-critical Q-curvatures.

This paper concerns the asymptotic behavior of zeros and critical points for monochromatic random waves φ λ of frequency λ on a compact, smooth, Riemannian manifold (M, g) as λ → ∞. We prove that the measure of integration over the zero set of φ λ restricted to balls of radius ≈ λ −1 converges in distribution to the measure of integration over the zero set of a frequency 1 random wave on R n , where n is the dimension of M . We also prove convergence of finite moments for the counting measure of the critical points of φ λ , again restricted to balls of radius ≈ λ −1 , to the corresponding moments for frequency 1 random waves. We then patch together these local results to obtain new global variance estimates on the volume of the zero set and numbers of critical points of φ λ on all of M. Our local results hold under conditions about the structure of geodesics on M that are generic in the space of all metrics on M , while our global results hold whenever (M, g) has no conjugate points (e.g is negatively curved).