Minimizing closed geodesics on polygons and disks

Abstract

We study $\frac{1}{k}$-geodesics, those closed geodesics that minimize on all subintervals of length $\frac{L}{k}$, where $L$ is the length of the geodesic. We develop new techniques to study the minimizing properties of these curves on doubled polygons, and demonstrate a sequence of doubled polygons where the minimizing index (the smallest $k$ such that the space admits a $\frac{1}{k}$-geodesic) is unbounded. We also compute the length of the shortest closed geodesic on doubled odd-gons and show that this length approaches $4\cdot$diameter.