Geometry and Topology Seminar

Let $(M^{n},g)$ be an $n$-dimensional Riemannian manifold and $1\leq k\leq n-1$. In the 1980's, Gromov studied the asymptotic behavior of the volumes $\omega^{k}_{p}(M^{n},g)$ of certain (possibly singular) $k$-dimensional minimal submanifolds $N^{k}_{p}$ of $(M^{n},g)$ arising from a Morse Theory of the volume functional (the Almgren-Pitts Min-Max Theory). He conjectured that these volumes $\omega_{p}^{k}(M^{n},g)$ behave when $p\to\infty$ similarly to the eigenvalues $\lambda_{p}$ of the Laplacian in $(M^{n},g)$, obeying the so-called Weyl Law for the Volume Spectrum. In 2016, Liokumovich, Marques and Neves solved the conjecture for $k=n-1$ and $n$ arbitrary. More recently, Guth and Liokumovich proved it for $1$-cycles in $3$-manifolds ($k=1$ and $n=3$). I will discuss the main difficulties which arise when the codimension $n-k$ of the cycles is greater than $1$, which boils down to proving parametric versions of the isoperimetric and coarea inequalities; and the applications of the Weyl law for $1$-cycles on generic density and equidistribution of stationary geodesic networks.