Extremal isosystolic metrics with multiple bands of crossing geodesics

Abstract

We apply recently developed convex programs to find the minimal-area Riemannian metric on $2n$-sided polygons ($n\geq 3$) with length conditions on curves joining opposite sides. We argue that the Riemannian extremal metric coincides with the conformal extremal metric on the regular $2n$-gon. The hexagon was considered by Calabi. The region covered by the maximal number $n$ of geodesics bands extends over most of the surface and exhibits positive curvature. As $n\to \infty$ the metric, away from the boundary, approaches the well-known round extremal metric on $\mathbb{RP}_2$. We extend Calabi's isosystolic variational principle to the case of regions with more than three bands of systolic geodesics. The extremal metric on $\mathbb{RP}_2$ is a stationary point of this functional applied to a surface with infinite number of systolic bands.