Extremal isosystolic metrics with multiple bands of crossing geodesics

IF 1 4区 物理与天体物理 Q3 PHYSICS, MATHEMATICAL
Usman Naseer, B. Zwiebach
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引用次数: 9

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.
具有多个交叉测地线带的极端等收缩度量
我们应用最近开发的凸规划来寻找$2n$边多边形($n\geq 3$)上的最小面积黎曼度量,曲线上的长度条件连接对边。我们认为黎曼极限度规与正则$2n$ -gon上的共形极限度规重合。六边形是卡拉比考虑的。测地线带的最大数目$n$所覆盖的区域扩展到大部分表面并呈现正曲率。当$n\to \infty$度规远离边界时,接近于$\mathbb{RP}_2$上众所周知的圆形极值度规。我们将Calabi的等收缩变分原理推广到具有三个以上收缩测地线带的区域。$\mathbb{RP}_2$上的极值度规是该函数应用于具有无限数量收缩带的表面的一个固定点。
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来源期刊
Advances in Theoretical and Mathematical Physics
Advances in Theoretical and Mathematical Physics 物理-物理:粒子与场物理
CiteScore
2.20
自引率
6.70%
发文量
0
审稿时长
>12 weeks
期刊介绍: Advances in Theoretical and Mathematical Physics is a bimonthly publication of the International Press, publishing papers on all areas in which theoretical physics and mathematics interact with each other.
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