Thurston's Asymmetric Metric

• Thurston's asymmetric metric is a non-symmetric Finsler metric defining the minimal Lipschitz constant required to deform hyperbolic structures in Teichmüller space.
• It employs both length-ratio and Lipschitz extremal formulations to study geodesics and stretch lines, offering insights into the dynamics of the mapping class group.
• The metric generalizes to diverse settings—including Euclidean triangles, flat metrics, and higher-rank representations—bridging intricate geometric and dynamical properties.

Thurston's asymmetric metric is a non-symmetric Finsler metric on Teichmüller space and related moduli spaces of geometric structures, encoding the minimal Lipschitz constant required to deform one structure into another in a fixed homotopy class. Originally formulated by W. P. Thurston in the context of the Teichmüller space of hyperbolic structures on a finite-type surface, the metric generalizes to a wide range of settings, including moduli of Euclidean triangles, triangulated surfaces, flat structures, semi-translation surfaces, and representations into higher-rank Lie groups. The metric underpins a rich metric geometry on moduli spaces, providing deep connections with length-spectrum rigidity, mapping class group dynamics, and polyhedral/convex geometry.

1. Fundamental Definitions and Formulations

Given an oriented surface $S$ of finite type (genus $g$ with $n$ punctures), the Teichmüller space $\mathcal{T}(S)$ consists of marked hyperbolic metrics up to isotopy. For $X,Y\in\mathcal{T}(S)$, Thurston's asymmetric metric is defined in two equivalent ways:

• Length-ratio (supremal) definition:

$$d_{\mathrm{Th}}(X,Y) = \log \sup_{\gamma \in \mathcal{S}} \frac{\ell_Y(\gamma)}{\ell_X(\gamma)}$$

where $\mathcal{S}$ is the set of free homotopy classes of essential simple closed curves (or measured laminations), and $\ell_X(\gamma)$ denotes the length of the geodesic representative in $X$.

• Lipschitz extremal definition:

$$d_{\mathrm{Th}}(X,Y) = \inf_{f \sim \mathrm{id}} \log \mathrm{Lip}(f)$$

where the infimum is over all homeomorphisms $f : X \rightarrow Y$ homotopic to the identity, and $\mathrm{Lip}(f)$ is the global Lipschitz constant of $f$.

Both definitions generalize to settings such as moduli of marked Euclidean triangles, unit-area flat metrics, and higher Teichmüller-type representation spaces (Saglam et al., 24 Apr 2025, Shi, 19 Oct 2025, Carvajales et al., 2022).

Key properties:

• $d_{\mathrm{Th}}(X,Y) \geq 0$ and $d_{\mathrm{Th}}(X,X) = 0$.
• $d_{\mathrm{Th}}$ generally fails to be symmetric: $d_{\mathrm{Th}}(X,Y)\neq d_{\mathrm{Th}}(Y,X)$.
• The triangle inequality holds.
• Nondegeneracy: $d_{\mathrm{Th}}(X,Y) = 0 \implies X=Y$, by marked-length-spectra rigidity (see also (Shi, 19 Oct 2025, Papadopoulos et al., 2012) for punctured surfaces and Fuchsian groups).

2. Infinitesimal and Finsler Geometry

Thurston’s metric induces a canonical (asymmetric) Finsler structure on $T_X \mathcal{T}(S)$, the tangent space at $X\in\mathcal{T}(S)$:

$$\|v\|_{X}^{\mathrm{Th}} = \sup_{\gamma \in \mathcal{S}} \frac{d\,\ell_\gamma(v)}{\ell_\gamma(X)}$$

for $v\in T_X \mathcal{T}(S)$, where $d\,\ell_\gamma$ is the differential of the length function at $X$ (Papadopoulos et al., 2011, Carvajales et al., 2022). This norm is positively homogeneous and convex but not symmetric. It is locally Lipschitz and induces the length structure underlying the Thurston metric.

For generalizations to spaces of marked triangles or triangulated surfaces, the Finsler norm takes the form

$F_X(v) = \max_{i} \frac{v_i}{A_i}\,,$

where $A_i$ are coordinate functions (e.g., side-length or triangle area data) (Saglam et al., 24 Apr 2025).

Explicit geodesics are constructed as stretch lines: integral curves of the Finsler norm where the supremum is realized by a single measured lamination, and their existence and uniqueness properties are critical for the global metric geometry (Théret, 2018, Théret, 2014, Dumas et al., 2016).

3. Geodesics, Stretch Lines, and Envelopes

Thurston geodesics need not be unique, reflecting the nonsymmetric character of the metric. The fundamental objects are "stretch lines," geodesics along which the optimal Lipschitz map stretches a directing lamination (or a geometric coordinate) by a uniform factor:

• On $(\mathcal{T}(S), d_{\mathrm{Th}})$, stretch lines are characterized via complete geodesic laminations, horocyclic foliations, and shear coordinates (Théret, 2018, Théret, 2014).
• On moduli of marked Euclidean triangles, geodesics are constructed by geometric interpolation in coordinate charts (Saglam et al., 24 Apr 2025).
• Stretch lines exhibit explicit exponential or log-linear scaling in the relevant structural coordinates.

Geodesic envelopes, the union of all geodesics between two points, exhibit uniform boundedness properties in low-complexity moduli (e.g., $S_{1,1}$ or $S_{0,4}$) (Bar-Natan, 2023). In general, concatenations of stretch segments along maximal laminations yield geodesics for the metric and encode the combinatorics of maximally-stretched laminations (Dumas et al., 2016). The local and global behavior of geodesics is intimately tied to the topology of the underlying surface and the combinatorial structure of the lamination complex.

4. Completeness and Topological Features

The (asymmetric) Thurston metric is, in general, forward but not backwards complete. However, in finite-dimensional cases such as the moduli of Euclidean triangles, it can be both forward and backward complete, with the metric completion corresponding exactly to the closure of the configuration space in the natural topology—no further boundary points arise (Saglam et al., 24 Apr 2025). For the classical Thurston metric, the metric completion requires projective measured laminations (the Thurston boundary) (Saglam et al., 24 Apr 2025, He et al., 3 Oct 2024).

Invariant balls ("left" and "right" balls) and the resultant topologies can differ for asymmetric metrics, especially in flat settings where the left-ball topology coincides with the geodesic-current topology and right-balls can be strictly coarser (Shi, 19 Oct 2025).

Completeness is established for symmetrized versions of the metric such as

$$d_{\max}(X,Y) = \max\{d_{\mathrm{Th}}(X,Y), d_{\mathrm{Th}}(Y,X)\}$$

and arithmetic means, which always yield complete metric spaces (Saglam et al., 24 Apr 2025).

5. Analogies and Generalizations

Thurston's asymmetric metric is part of a broader geometric framework:

• Hyperbolic vs. Euclidean/Flat Settings: In the space of marked Euclidean triangles or flat metrics, analogues of Thurston's metric are defined using appropriate length functions or edge-coordinates, yielding parallel but often simpler metric structures (e.g., global completeness, finiteness) (Saglam et al., 24 Apr 2025, Shi, 19 Oct 2025).
• Semi-translation Surfaces: Thurston-type asymmetric metrics are constructed, with equality between extremal Lipschitz and supremum-length definitions hinging on combinatorial polygon-retraction lemmas (Wolenski, 2018).
• Representation Varieties: For Anosov representations and Margulis spacetimes, the metric is extended by using length and entropy data attached to representations, and the associated Finsler norms are explicitly computed in key examples such as Hitchin components and convex projective structures (Carvajales et al., 2022, Gongopadhyay et al., 8 Dec 2025).
• Action of Mapping Class Group: The metric provides a translation length for mapping classes, giving rise to an analogue of the Nielsen–Thurston classification (elliptic, parabolic, hyperbolic, pseudo-hyperbolic) based on optimal stretch (Liu et al., 2011).
• Finsler Dualities and Earthquake Metric: Infinitesimal structures dualize to notable geometric norms and relate closely to the earthquake metric, which provides a distinct but related asymmetric Finsler geometry on Teichmüller space (Huang et al., 30 Apr 2024).

Table: Ratio and Lipschitz Definitions in Different Settings

Setting	Ratio Definition	Lipschitz Definition
Hyperbolic surface	$$\log\sup_\gamma \frac{\ell_Y(\gamma)}{\ell_X(\gamma)}$$	$\inf_{f \sim \mathrm{id}} \log\mathrm{Lip}(f)$
Marked Euclidean triangles	$\log\max_i \frac{A'_i}{A_i}$	$\inf_{f:\, BX\to BY} \log \mathrm{Lip}(f)$
Flat metrics (half-translation)	$$\log\sup_\gamma \frac{\ell_{q_2}(\gamma)}{\ell_{q_1}(\gamma)}$$	as above, now for maps between flat surfaces
Anosov representations	$$\log\sup_{[\gamma]} \frac{h_\sigma L_\sigma([\gamma])}{h_\rho L_\rho([\gamma])}$$	as above, with entropy normalization
Semi-translation surfaces	$$\sup_{\text{saddle connections}} \log\frac{\ell_{q_2}}{\ell_{q_1}}$$	$\inf_{\varphi} \log \mathrm{Lip}(\varphi)$

6. Convexity, Polyhedral Geometry, and Functional Properties

Convexity in shear coordinates is a central feature: the length of a measured lamination is a convex function in Thurston's shear (length) coordinates. If the lamination is transverse to all leaves of a fixed complete lamination, this convexity is strict (Théret, 2014). The supremum of convex log-length functions imparts a convex unit ball in the (asymmetric) Finsler norm. This underlies both generic uniqueness and polyhedrality of the Finsler norm (Dumas et al., 2016).

The extremal directions (stretch vectors) are precisely the tangent directions for which the supremum in the norm is achieved by a single lamination, and each tangent sphere is a convex, polyhedral (non-symmetric) body whose combinatorics encode the length spectrum (Bar-Natan, 2023, Dumas et al., 2016).

7. Extensions and Open Problems

Thurston's metric admits further extensions and open directions:

• The earthquake metric is another asymmetric Finsler geometry, distinct in its geodesic structure and completion theory, and is not forward-complete; FD-completion yields natural compactifications (Huang et al., 30 Apr 2024).
• On flat and semi-translation strata, existence and properties of extremal Lipschitz maps reduce to subtle combinatorics of polygons, with open conjectures about the construction of 1-Lipschitz polygonal retractions (Wolenski, 2018).
• In higher-rank representation theory, the metric structure captures dynamical and spectral data, and entropy normalization is essential for genuine asymmetry (Gongopadhyay et al., 8 Dec 2025, Carvajales et al., 2022).
• For moduli of surfaces with boundary, generalizations involve arc-length maximization and explicit geodesic constructions (Alessandrini et al., 2019).

Recent research continues to elaborate the metric theory in both classical and flat metric contexts, explore the relationship with other moduli metric structures (Weil–Petersson, Teichmüller metric), and clarify the role of non-symmetry, completeness, and dynamics in the large-scale structure of moduli spaces (Dumas et al., 2016, Shi, 19 Oct 2025, Bar-Natan, 2023, Saglam et al., 24 Apr 2025, Huang et al., 30 Apr 2024).