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# Advances in Theoretical and Mathematical Physics

## Volume 26 (2022)

### Number 5

### Extremal isosystolic metrics with multiple bands of crossing geodesics

Pages: 1273 – 1346

DOI: https://dx.doi.org/10.4310/ATMP.2022.v26.n5.a7

#### Authors

#### 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.

Published 30 March 2023