Ambient geometry via min-max widths of embedded circles

Abstract

We prove a lower bound for the Birkhoff min-max invariant of a Riemannian sphere in terms of the min-max width of its embedded circles. The main tool is a method to induce a sweepout by pairs of points in an embedded circle from a given sweepout of the sphere by closed curves, so that the two points of each pair of the induced sweepout lie in nearby curves of the ambient sweepout. Moreover, considering a non-compact complete Riemannian manifold, we relate upper bounds for the min-max width of its embedded circles to the existence of continuous functions from the manifold to finite graphs with level sets that have uniformly bounded diameters, thus giving an estimate for its $1$-width in Urysohn's quantitative dimension theory. In the specific case of the Euclidean plane, we bound from below the classical width of a closed simple curve by its min-max width. Finally, we prove related rigidity results characterizing the round spheres among Zoll spheres.