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Finsler metrics on $1/n$-translation structures on surfaces

Beatrice Pozzetti, Jiajun Shi

Abstract

We define compatible Finsler distances on $1/n$-translation surfaces, we study their geodesics, and construct a Liouville current for each such metric, that is a geodesic current that encodes the information of the length of the closed curves. The construction is based on multi-foliations, a generalization of measured foliations of independent interest.

Finsler metrics on $1/n$-translation structures on surfaces

Abstract

We define compatible Finsler distances on -translation surfaces, we study their geodesics, and construct a Liouville current for each such metric, that is a geodesic current that encodes the information of the length of the closed curves. The construction is based on multi-foliations, a generalization of measured foliations of independent interest.

Paper Structure

This paper contains 21 sections, 56 theorems, 39 equations, 18 figures.

Table of Contents

  1. Introduction
  2. Singular Finsler metrics on $1/n$-translation surfaces
  3. The $1/n$-translation surfaces and their CAT(0) metric
  4. Finsler metrics
  5. CAT(0) geodesics and Finsler metrics
  6. Straight geodesics
  7. $w$-brushes
  8. Supporting vectors
  9. The projection along a brush and the proof of Theorem \ref{['thmA']}
  10. Multi-foliations and associated geodesic currents
  11. Multi-foliations
  12. Leaves of the multi-foliation and small boxes
  13. The multi-foliation current $\mu_\theta$
  14. Geometric properties of multi-foliation currents
  15. Positive $k$-crossing
  16. ...and 6 more sections

Key Result

Theorem 1.1

For any norm $\lVert\cdot\rVert$ on $\mathbb{R}^2$ invariant by rotations of order $n$, and every $1/n$-translation structure $\Sigma$, the piecewise-straight paths are geodesics for $d^F$ and realize the smallest length in (relative) homotopy classes. $\blacktriangleleft$$\blacktriangleleft$

Figures (18)

  • Figure 1: A $\frac{1}{4}$-translation surface on the left and a $\frac{1}{3}$ translation surface on the right. Sides with the same label are identified isometrically. The surface on the left has one cone point of angle $6\pi$, the surface on the right has two cone points of angles $4\pi$, both surfaces have genus two.
  • Figure 2: If one angle formed by $\gamma_l,\gamma_r$ is less than $\pi$, then the length of red curve is smaller than the sum of the length of $\gamma_l,\gamma_r$
  • Figure 3: The black vectors form the 3-web at angle 0, and the 4-web at angle 0, and the blue polygons are the unit spheres of the associated norms.
  • Figure 4: A local length minimizer for the $\ell^1$-metric in $\mathbb{R}^2$ in blue, which is not a Finsler geodesic.
  • Figure 5: A left-turning straight $\mathop{\mathrm{CAT}}\nolimits(0)$ geodesic in red, and a right-turning straight $\mathop{\mathrm{CAT}}\nolimits(0)$ geodesic in blue sharing a saddle connection.
  • ...and 13 more figures

Theorems & Definitions (130)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3: See duchin2010length for $n=2$
  • proof
  • Proposition 2.4
  • Proposition 2.5
  • proof
  • ...and 120 more