Scaling limit for the kernel of the spectral projector and remainder estimates in the pointwise Weyl law

Abstract

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