Event Details
Abstract: On the equidistribution of closed geodesics and geodesic nets
Given a Riemannian manifold $(M^n, g)$, it is natural to ask whether it admits any closed geodesics and if so, how many are there and how are they distributed along the manifold.
In this talk, I will present the following result: given a closed 2-manifold $M^2$, for a generic (in the Baire sense) Riemannian metric $g$ on $M^2$ there exists an equidistributed sequence $\{\gamma_i\}_{i \in \mathbb{N}}$ of closed geodesics on $(M^2, g)$.
The same question in a higher dimensional ambient manifold $M^n$ turns out to be much harder and is still widely open. However, we can approach the same problem but considering stationary geodesic nets (which are embedded graphs in $(M, g)$ which are stationary with respect to the length functional) instead of closed geodesics.
Together with Yevgeny Liokumovich, we showed that for a Baire-generic metric $g$ on a fixed closed manifold $M^n$, the union of all embedded stationary geodesic nets on $(M^n, g)$ is dense in $M^n$. I will also present the following stronger result: if $n \geq 3$, given a closed $n$-manifold $M^n$, for a Baire-generic metric $g$ on $M^n$ there exists an equidistributed sequence $\{\gamma_i\}_{i \in \mathbb{N}}$ of stationary geodesic nets on $(M^n, g)$. For $n = 3$, this was joint work with Xinze Li.
