Low min-max widths of the unit disk

Abstract

For 2-manifolds with non-empty convex boundary, we prove a regularity result for one dimensional stationary varifolds V with free boundary, such that are Z2-almost minimizing in annuli. That regularity says that V is a free boundary finite geodesic network. Using that regularity we can deduce some properties of V, as an estimate for the one-dimensional density. Together with the Min-Max Theorem and the p-sweepouts given by real algebraic varieties, we were able to calculate the first p-widths of the unit ball B2 and of planar full ellipses close to B2. Those p-sweepouts are not always optimal. However, in our situations they are almost optimal.