Thurston's asymmetric metric for the space of flat metrics

TL;DR

The paper extends Thurston's asymmetric metric from hyperbolic metrics to the space $Flat(S)$ of unit-area flat metrics arising from half-translation structures. It defines $K(q_1,q_2)=\log\sup_{\gamma\in\mathcal{S}}\frac{\ell_{q_2}(\gamma)}{\ell_{q_1}(\gamma)}$ using simple closed curves and proves nondegeneracy: $K(q_1,q_2)>0$ for $q_1\neq q_2$, while also analyzing two asymmetric topologies induced by the metric and comparing them to the geodesic-current topology. The proof strategy combines cylinder-curve analysis, Liouville current intersections, Dehn twists, and $SL_2\mathbb{R}$-orbit arguments to show rigidity: if $\ell_{q_2}(\gamma)\ge\ell_{q_1}(\gamma)$ for all simple curves, then the metrics must coincide up to $SL_2\mathbb{R}$-equivalence, hence are identical. Additionally, the work clarifies the topological structure on $Flat(S)$, showing $B^l_q(R)$ aligns with the current topology, while $B^r_q(R)$ generally yields a different topology and can fail to be proper under degeneration to mixed structures.

Abstract

Thurston introduced in his seminal work an asymmetric metric on Teichmüller space by the ratio of simple closed curve length. In this paper, we generalize the idea and define an asymmetric metric on the space of unit-area flat metrics coming from half-translation structures on a closed surface. We also discuss two different topologies coming from the asymmetry.

Figures (6)

• Figure 1: Two different cases of $T_\alpha\beta$ (black curve) when $i(\alpha,\beta)=1$. It depends on the intersection angle whether the blacks curves are geodesics. On the left $\alpha$ is a cylinder curve, the black curve is not a geodesic and $\ell(T_\alpha\beta)<\ell(\alpha)+\ell(\beta)$, while on the right it is and $\ell(T_\alpha\beta)=\ell(\alpha)+\ell(\beta)$.
• Figure 2: The surface on the right is obtained by identifying parallel sides of the polygon on the left. All labeled sides in the left picture are oriented from left to right. The vertices of the polygon are glued to two cone points.
• Figure 3: A pair of pants with cylinder boundary curves $a,b,c$. Glue pairs of sides with the same label in the polygon, where I, II are glued in the same orientation and III, IV are glued in the opposite orientation. The red curve is the geodesic representative of $a*b^{-1}$.
• Figure 4: Left: nontrivial intersection at a cone point, mared angles no less than $\pi$. Middle: nontrivial intersection sharing a saddle conenction with the same orientation, marked angles automatically no less than $\pi$ as they are geodesics. Right: nontrivial intersection sharing a saddle connection with different orientation.
• Figure 5: New arc $\beta'$ after the twist in three cases. The left one and the middle one are geodesics, while the right one is not.

Theorems & Definitions (62)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Definition 2.1
• Definition 2.2
• Lemma 2.3: strebel1966quadratische
• Proposition 2.4: masur2002rational
• Definition 2.5
• Definition 2.6
• Theorem 2.7: bonahon1988geometry