Ambient geometry via min-max widths of embedded circles
arXiv:2607.12979
The paper establishes bounds for min‑max invariants of Riemannian spheres using the widths of embedded circles, introduces a technique to derive sweepouts of point pairs from curve sweepouts, connects these widths to functions onto finite graphs for estimating 1‑width, and proves rigidity results characterizing round spheres among Zoll spheres.
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 points of each pair of the induced sweepout lie close to two curves of the ambient sweepout that are close to each other. 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 simple closed curve by its min-max width. Finally, we prove related rigidity results characterizing the round spheres among Zoll spheres.
32 pages, 1 figure. Minor revision, references added. Submitted version