Obstructions and a Liouville-domination metric on a proper universal locus

Definitions and mathematical statement

The actual question attributed to D. Dumas is whether one can define a Thurston metric on universal Teichmüller space. The following section heading about infinitely generated first-kind Fuchsian groups and its introductory sentence are record-boundary contamination, not hypotheses of Problem 3.9. Use the model \[ \mathcal T_{\rm univ}=\operatorname{Mob}^+(S^1)\backslash\operatorname{QS}^+(S^1), \] where equivalence is postcomposition by a Möbius transformation. The source does not state which axioms, finiteness, topology, geodesics or compatibility with finite-type Thurston metrics are required.

The supplied aid proposes a metric on a proper sublocus. Let $\mathcal G$ be the space of unoriented geodesics of $\mathbb H^2$, identified with distinct unordered boundary pairs, and let $L_0$ be its standard Liouville current. For a quasisymmetric boundary map $f$, let $L_f=f^*L_0$; this is unchanged under postcomposition by Möbius maps. Define \[ \mathcal U=\{[f]:\ \exists\,0<c_f\le C_f<\infty,\quad c_fL_0\le L_f\le C_fL_0\}. \] The inequalities are domination of measures, not bounds on finitely many selected boxes. On this locus set \[ d_{\rm LD}([f],[g])=\log\operatorname*{ess\,sup}_{L_0} \frac{{\,\mathrm{d}} L_g/{\,\mathrm{d}} L_0}{{\,\mathrm{d}} L_f/{\,\mathrm{d}} L_0}. \] The aid reports that this is a finite directed metric: it is nonnegative, separates equivalence classes, and satisfies the triangle inequality. For the stated Liouville normalization, complementary boxes $Q,Q^\perp$ determined by four cyclic boundary points obey \[ e^{-L_f(Q)}+e^{-L_f(Q^\perp)}=1. \] This rules out strict one-sided domination with constant below one and forces equality of currents in the zero-distance case; equality of pullback currents determines the boundary map up to Möbius postcomposition. The Radon--Nikodym quotient chain rule supplies the triangle inequality. The report stresses that $\mathcal U$ is proper: quasisymmetric power maps $x\mapsto\operatorname{sgn}(x)|x|^a$ on the real boundary, $a>0$, $a\ne1$, supply failures of uniform comparison. Thus the construction is not a finite metric on all of universal Teichmüller space.

Two literal transplants are also obstructed. The disk has no essential closed-curve length family. Moreover unrestricted homeomorphic boundary extensions do not give a useful forward Lipschitz constant. In hyperbolic polar coordinates, for $0<a\le1$, \[ R_a(r,\theta)=(ar,\theta) \] is a disk homeomorphism extending the identity boundary map, with hyperbolic Lipschitz constant at most $a$. Hence the infimum is zero already for the identity boundary datum; composing with a finite-Lipschitz reference extension gives the same collapse whenever such an extension is available. Taking its logarithm cannot produce a normalized distance. These unrestricted maps need not belong to a more tightly controlled extension class. The aid's proper-locus metric and these failures leave the general definition and desired compatibility properties unresolved.

Short English statement

Seek an appropriate universal Thurston metric after removing the next section's text from the record. The supplied aid defines a finite Liouville-domination metric only on uniformly comparable currents and explains why empty closed-curve data and unrestricted forward-Lipschitz extensions do not supply a global answer.

Sources

• D. Dumas, Problem 3.9, AIM workshop Lipschitz metric on Teichmueller space, https://aimath.org/pastworkshops/lipschitzteichproblems.pdf. The supplied verification identifies the exact boundary on PDF page 4 and excludes the subsequent first-kind Fuchsian-group section.
• Local UnsolvedMath registry, permanent ID 20003112, exact code catalogue entry; canonical location aim-topology-notes.json, notes[199], dated 14 August 2026; statement labeled exact although its stored text includes the explicitly separated contamination.
• Research attempt 1: partial result, problem status unknown at the run, rigor label proved, novelty confidence low; registry status partially solved. The comparable-current construction and radial/power-map tests are machine-generated research aids, not peer reviewed; independent verification is required before citation. No separate authoritative bibliography is supplied.