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

Usman Naseer (Department of Physics, Harvard University, Cambridge, Massachusetts, U.S.A.; Department of Physics and Astronomy, Uppsala University, Uppsala, Sweden; and Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge Mass., U.S.A.)

Barton Zwiebach (Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge Mass., U.S.A.)

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.

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Published 30 March 2023