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Bruno Staffa

Publications and source records attributed to Bruno Staffa.

5 recordsLinked to original sources
arXiv2024

Weyl law for 1-cycles

We prove the Weyl law for the volume spectrum for $1$-cycles in $n$-dimensional manifolds which was conjectured by Gromov. We follow the strategy of Guth and Liokumovich of obtaining the Weyl law from parametric versions of the coarea inequality and the isoperimetric inequality. A version of the later for families of $0$-cycles is shown in this article. We also obtain approximation results by $\delta$-localized families, which are used to prove the parametric inequalities.

math.DG↗
arXiv2022

On the equidistribution of closed geodesics and geodesic nets

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.

math.DG↗
arXiv2021

Generic density of geodesic nets

We prove that for a Baire-generic Riemannian metric on a closed smooth manifold, the union of the images of all stationary geodesic nets forms a dense set.

math.DG↗
arXiv2021

Bumpy Metrics Theorem for Geodesic Nets

Stationary geodesic networks are the analogs of closed geodesics whose domain is a graph instead of a circle. We prove that for a Baire-generic Riemannian metric on a smooth manifold $M$, all connected embedded stationary geodesic nets are non-degenerate.

math.DG↗