Isosystolic extremal metrics with multiple bands of crossing geodesics
Authors: Usman Naseer, Barton Zwiebach
Preprint number: UUITP-8/19
We apply recently developed convex programs to find the minimal-area Riemannian metric on 2n-sided polygons (n≥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 case n=3 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→∞ the metric, away from the boundary, approaches the well-known round extremal metric on RP2. We extend Calabi's isosystolic variational principle to the case of regions with more than three bands of systolic geodesics. The extremal metric on RP2 is a stationary point of this functional applied to a surface with infinite number of systolic bands.