Logarithmic improvements in the Weyl law and exponential bounds on the number of closed geodesics are predominant

TL;DR

This paper introduces the concept of predominance for Riemannian metrics on compact manifolds, demonstrating that for such metrics, the number of closed geodesics grows at most stretched exponentially, and the Weyl law error term can be improved logarithmically.

Contribution

It establishes that predominant metrics have specific bounds on closed geodesics and Weyl law remainders, linking geometric properties to measure-theoretic notions.

Findings

Number of closed geodesics has a stretched exponential upper bound.

Weyl law error term can be improved by a power of log λ.

Predominant metrics are dense and have non-degenerate nearly closed orbits.

Abstract

Let $M$ be a smooth compact manifold of dimension $d$ without boundary. We introduce the concept of predominance for Riemannian metrics on $M$, a notion analogous to full Lebesgue measure which, in particular, implies density. We show that for a predominant metric, the number of closed geodesics of length smaller than $T$ has a stretched exponential upper bound in $T$. In addition, we study remainders in the Weyl law for predominant metrics. The Weyl law states that the number of Laplace-Beltrami eigenvalues smaller than $\lambda^2$ is asymptotic to $C\lambda^d$ with an $O(\lambda^{d-1})$ error. We show that, for a predominant metric, the estimate on the error can by improved by a power of $\log \lambda$. After an application of recent results of the authors in the case of the Weyl law, these estimates follow from a study of the non-degeneracy properties of nearly closed orbits for predominant sets of metrics.