The first width of non-negatively curved surfaces with convex boundary
In this paper, free boundary geodesic networks whose length realize the first min-max width of the length functional are investigated. This functional acts on the space of relative flat 1-dimensional cycles modulo 2 in a compact surface with boundary. The widths are special critical values of the volume functional in some class of submanifolds which naturally arise in the Min-max Theory of Almgren and Pitts. The main result of this work concerns the existence of a geodesic network with a rather simple structure which realizes the first width of a surface with non-negative sectional curvature and strictly convex boundary. More precisely, it is either a simple geodesic meeting the boundary orthogonally, or a geodesic loop with vertex at a boundary point determining two equal angles with that boundary curve.