Riesz means of the counting function of the Laplace operator on compact manifolds of non-positive curvature

  • Kamil Mroz

    Loughborough University, UK
  • Alexander Strohmaier

    Loughborough University, UK

Abstract

Let (M,g)(M, g) be a compact, dd-dimensional Riemannian manifold without boundary. Suppose further that (M,g)(M,g) is either two dimensional and has no conjugate points or (M,g)(M,g) has non-positive sectional curvature. The goal of this note is to show that the long time parametrix obtained for such manifolds by Bérard can be used to prove a logarithmic improvement for the remainder term of the Riesz means of the counting function of the Laplace operator.

Cite this article

Kamil Mroz, Alexander Strohmaier, Riesz means of the counting function of the Laplace operator on compact manifolds of non-positive curvature. J. Spectr. Theory 6 (2016), no. 3, pp. 629–642

DOI 10.4171/JST/134