Weyl law for 1-cycles (2410.23192v1)

Abstract: 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.