On the equidistribution of closed geodesics and geodesic nets

TL;DR

This paper proves that for a generic set of Riemannian metrics on closed manifolds, there exist sequences of closed geodesics or geodesic nets that become uniformly distributed across the manifold, extending previous results to higher dimensions.

Contribution

It establishes the existence of equidistributed geodesic sequences and nets in higher dimensions under generic metrics, generalizing prior two- and three-dimensional results using the Weyl Law.

Findings

Existence of equidistributed closed geodesics on 2-manifolds

Existence of equidistributed embedded stationary geodesic nets on 3-manifolds

Potential generalization to higher dimensions assuming Weyl Law holds

Abstract

We show that given a closed $n$-manifold $M$, for a generic set of Riemannian metrics $g$ on $M$ there exists a sequence of closed geodesics that are equidistributed in $M$ if $n=2$; and an equidistributed sequence of embedded stationary geodesic nets if $n=3$. One of the main tools that we use is the Weyl Law for the volume spectrum for $1$-cycles, proved by Liokumovich, Marques and Neves for $n=2$ and more recently by Guth and Liokumovich for $n=3$. We show that our proof of the equidistribution of geodesic nets can be generalized for any dimension $n\geq 2$ provided the Weyl Law for $1$-cycles in $n$-manifolds holds.