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 ≥ 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 → ∞ the metric, away from the boundary, approaches the well-known round extremal metric on 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 RP 2 is a stationary point of this functional applied to a surface with infinite number of systolic bands.

Hyperbolic three-string vertex

“…Now let v(z) be a vector field holomorphic everywhere except for the punctures. 9 That is, it changes as v(z) → ṽ(z) = (∂ z)v(z) under z → z. Note that v(z) should be regular at z = ∞ by its definition, so we must ensure z −2 v(z) is finite as z → ∞ by the inversion map z → z = 1/z.…”

“…The minimal area metrics for higher genus surfaces, however, are not known explicitly and still lack rigorous proof of existence. Nonetheless, one may expect that these will soon follow in the light of the recent discoveries [7][8][9].…”

Sharp upper bounds on the length of the shortest closed geodesic on complete punctured spheres of finite area

“…It was later proved that this structure is common to all extremal nonpositively curved surfaces; see [26]. Extremal systolic inequalities in a fixed conformal class have been investigated in relation with closed string field theory; see [18,19,30] for the most recent contributions and the references therein.…”

Sharp upper bounds on the length of the shortest closed geodesic on complete punctured spheres of finite area

“…It was later proved that this structure is common to all extremal nonpositively curved surfaces; see [26]. Extremal systolic inequalities in a fixed conformal class have been investigated in relation with closed string field theory; see [18], [19], [28] for the most recent contributions and the references therein.…”