Explicit bounds on eigenfunctions and spectral functions on manifolds hyperbolic near a point

TL;DR

This paper provides explicit local bounds on eigenfunctions, spectral functions, and heat traces on hyperbolic manifolds, with applications to discrete spectra and local geometric analysis.

Contribution

It introduces new explicit bounds for the remainder in the local Weyl law and derivatives, applicable to locally hyperbolic manifolds and near points with hyperbolic metrics.

Findings

Explicit bounds for the local Weyl law remainder

Estimates for derivatives of the remainder term

Bounds for C^k-norms of eigenfunctions and local heat trace

Abstract

We derive explicit bounds for the remainder term in the local Weyl law for locally hyperbolic manifolds, we also give the estimates of the derivative of this remainder. We use these to obtain explicit bounds for the C^k-norms of the L^2-normalised eigenfunctions in the case spectrum of the Laplacian is discrete, e.g. for closed Riemannian manifolds. We also derive bounds for the local heat trace. Our estimates are purely local and therefore also hold for any manifold at points near which the metric is locally hyperbolic.