A lower bound for the $\Theta$ function on manifolds without conjugate points

TL;DR

This paper proves that the $ heta$ function on manifolds without conjugate points is uniformly bounded below, extending Green's 2D result and confirming the Bérard remainder in Weyl law for such manifolds across all dimensions.

Contribution

It establishes a lower bound for the $ heta$ function on manifolds without conjugate points, generalizing previous two-dimensional results to higher dimensions.

Findings

The $ heta$ function is uniformly bounded from below on these manifolds.

The Bérard remainder in Weyl law applies without dimension restrictions.

The result simplifies understanding spectral properties of manifolds without conjugate points.

Abstract

In this short note, we prove that the usual $\Theta$ function on a Riemannian manifold without conjugate points is uniformly bounded from below. This extends a result of Green in two dimensions. This elementary lemma implies that the B\'erard remainder in the Weyl law is valid for a manifold without conjugate points, without any restriction on the dimension.