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

2022/04/25 by Yaiza Canzani, Jeffrey Galkowski, Canzani, Yaiza +1 · 1 citation
Mathematics · #Analysis of PDEs (math.AP) #Analytic and geometric function theory #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Mathematical Physics (math-ph) #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2204.11921

openalex publication_date 2022/04/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

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 λ² is asymptotic to Cλ^d with an O(λ^d-1) error. We show that, for a predominant metric, the estimate on the error can by improved by a power of log λ. 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.

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