Boundary-sensitive extensions of the Thurston metric

Definitions and mathematical statement

The source asks whether the Thurston metric can be defined on the non-reduced Teichmüller space of a hyperbolic surface with ideal boundary. The admissible surface class, meaning of ideal boundary, boundary marking equivalence, and desired normalization are not specified. Write $\pi:\mathcal T^{\mathrm{nr}}\to\mathcal T^{\mathrm{red}}$ for the quotient forgetting the additional boundary marking, in the intended reduced/non-reduced conventions.

The aid first reports that ordinary closed-curve lengths and free-endpoint arc lengths factor through $\pi$. Thus any distance defined solely from those spectra is constant on each boundary-marking fiber and cannot separate non-reduced points in a nontrivial fiber. This is an obstruction for those particular observables, not for arcs with fixed individually marked ideal endpoints.

A second obstruction concerns the raw infimal-Lipschitz formula if its admissible relative maps include all boundary-fixing Lipschitz homeomorphisms. In hyperbolic polar coordinates on the Poincaré disk, the radial map \[ h_c(r,\theta)=(cr,\theta),\qquad0<c<1, \] where $r$ is hyperbolic distance from the origin, fixes the ideal boundary pointwise and has hyperbolic Lipschitz constant $c$. Hence the infimum of these constants is zero even for the disk self-comparison, and its unnormalized logarithm collapses to $-\infty$. These maps are not asserted quasiconformal with a finite global dilatation; restricting the admissible maps to quasiconformal maps would require a separate analysis. The failure pertains to the stated raw formula and class of maps.

For boundary observables, take a hyperbolic $0$-metric with boundary defining function $\rho$, and choose ideal-endpoint geodesics between distinct marked boundary points $p_i$. A renormalized length has the form \[ L_\rho(p_i,p_j)=\lim_{\varepsilon\downarrow0} \left(\operatorname{length}(\gamma_{ij}\cap\{\rho\geq\varepsilon\})+2\log\varepsilon\right), \] when that limit and cutoff convention apply. Under $\rho\mapsto e^\omega\rho$, its change in this convention is $\omega(p_i)+\omega(p_j)$. The aid's balance criterion for a linear combination is therefore \[ \sum_{i<j}c_{ij}L_\rho(p_i,p_j)\text{ is boundary-gauge independent} \Longleftrightarrow \sum_{j\ne i}c_{ij}=0\quad\text{for every endpoint }i, \] with symmetric coefficients $c_{ij}=c_{ji}$. In the model with at most one chosen length for each distinct unordered endpoint pair, there is no nonzero balanced coefficient system on at most three endpoints over ${\mathbb{R}}$; four endpoints admit alternating combinations such as \[ L_{12}+L_{34}-L_{14}-L_{23}, \] which give boundary cross-ratio observables in the appropriate hyperbolic convention. If multiple geodesic classes with the same endpoints are allowed, differences of their lengths already cancel the endpoint gauge, so the four-endpoint minimality statement requires this pair-indexed qualification. The precise $0$-metric hypotheses, geodesic choices, and cross-ratio normalization are not supplied. Gauge independence alone is not a distance construction.

Finally, wherever a finite asymmetric arc distance $d_A$ is defined on the reduced space and the usual non-reduced Teichmüller distance $d_T^{\mathrm{nr}}$ is finite, the aid gives the regularization \[ D_\epsilon(X,Y)=d_A(\pi X,\pi Y)+\epsilon d_T^{\mathrm{nr}}(X,Y),\qquad\epsilon>0. \] With the usual nonnegativity and triangle inequalities for these inputs, the positive Teichmüller term makes this an honest separating asymmetric metric. Its $\epsilon$-dependence and use of the preexisting Teichmüller metric are retained. The aid describes a canonical pure boundary-sensitive Thurston construction as still open; this regularization does not by itself provide it.

Short English statement

Ordinary length spectra lose the extra boundary marking, and the stated raw Lipschitz formula can collapse if all boundary-fixing radial maps are allowed. Balanced renormalized lengths provide gauge-free boundary data, while adding a positive non-reduced Teichmüller term gives a regularized metric. The canonical pure construction remains open according to the aid.

Sources

• Local UnsolvedMath registry, permanent ID $20003111$, exact source code catalogue entry; canonical aim-topology-notes.json, notes[198], tag problem.
• AIM workshop "Lipschitz metric on Teichmueller space," Problem 3.8: https://aimath.org/pastworkshops/lipschitzteichproblems.pdf. The supplied verification identifies this official-list question as exact. The PDF is cited without being fetched.
• The original literature field is empty. No authoritative bibliographic reference for the boundary-marking conventions, hyperbolic $0$-metric renormalization, or proposed metric construction is supplied.
• Research attempt 1: result partial_result, problem status at run open, rigor label proved, novelty confidence low; classification PARTIAL-PROGRESS, registry status partially_solved. Created and updated 14 August 2026. These are supplied assessments, not acceptance of a canonical Thurston extension.
• Attribution: Ulam AI / UnsolvedMath Contributors, CC BY 4.0. The machine-generated research aid is not peer reviewed and requires independent verification before citation.