The one-sided length-spectrum obstruction and a regularized asymmetric metric

Definitions and mathematical statement

Let $S_0=\mathbb H^2/\Gamma_0$ be the intended reference hyperbolic surface and $\mathcal T_{\mathrm{qc}}(\Gamma_0)$ its quasiconformal Teichmüller space of marked structures. The source asks whether a Thurston metric can be defined there, but supplies neither the hypotheses on $\Gamma_0$ nor the meaning of extending the finite-type metric. Fix the intended nonempty family $\mathcal C$ of marked simple curve classes of positive hyperbolic length, with boundary/peripheral conventions required for marked-length rigidity. The following construction requires that these length functions separate points; otherwise it produces only a pseudometric before taking the length-spectrum quotient.

For $X,Y\in\mathcal T_{\mathrm{qc}}(\Gamma_0)$ define \[ A(X,Y)=\log\sup_{\alpha\in\mathcal C}\frac{\ell_Y(\alpha)}{\ell_X(\alpha)},\qquad \Delta(X,Y)=\max\{A(X,Y),A(Y,X)\}. \] Quasiconformal length distortion makes both directed quantities finite. One has $A(X,X)=0$ and the directed triangle inequality, but these do not alone give nonnegativity and separation. With the strong asymmetric-metric convention $d(X,Y)>0$ for $X\ne Y$, the naive formula $A$ is an asymmetric metric exactly when strict marked-length non-domination holds: \[ \forall X\ne Y,\qquad \sup_{\alpha\in\mathcal C}\frac{\ell_Y(\alpha)}{\ell_X(\alpha)}>1, \] equivalently some $\alpha$ satisfies $\ell_Y(\alpha)>\ell_X(\alpha)$, for every ordered pair. This differs from the weaker separation convention requiring only that $d(X,Y)=d(Y,X)=0$ imply $X=Y$.

The aid's regularization avoids assuming that unresolved condition. In the convention without a factor of one half, $\Delta=\sup_\alpha|\log(\ell_Y(\alpha)/\ell_X(\alpha))|$ is the symmetric length-spectrum metric, under the stipulated rigidity hypothesis. Since $-\Delta\leq A\leq\Delta$, for any real $c>1$ put \[ D_c(X,Y)=A(X,Y)+c\Delta(X,Y). \] Then $D_c$ is a possibly asymmetric metric satisfying \[ (c-1)\Delta\leq D_c\leq(c+1)\Delta, \qquad\max\{D_c(X,Y),D_c(Y,X)\}=(c+1)\Delta(X,Y), \] \[ D_c(X,Y)-D_c(Y,X)=A(X,Y)-A(Y,X). \] Its triangle inequality follows by adding those for $A$ and $\Delta$, and the lower bound gives strong positivity off the diagonal. Thus it has exactly the same directional defect as $A$ and an explicitly controlled symmetrization. If the chosen length family does not separate the quasiconformal markings, the same inequalities give a pseudometric and do not repair that defect.

The construction is an asymmetric length-spectrum regularization. It is not asserted to coincide with an infimal Lipschitz metric, preserve the finite-type Thurston formula, be geodesic, be canonical independently of $c$, or induce the quasiconformal topology. Deciding the non-domination condition and the desired additional meaning of a Thurston metric remains outside the supplied partial result.

Short English statement

The naive one-sided length formula is finite and satisfies a directed triangle inequality, but being a genuine asymmetric metric requires a strict non-domination property for every ordered pair. The supplied regularization adds a multiple of the symmetric length-spectrum metric and gives explicit bounds and symmetrization. This constructs a controlled metric when marked lengths separate points, without establishing that it is the intended Thurston extension.

Sources

• Local registry: permanent catalogue entry, exact source code catalogue entry; aim-topology-notes.json, notes[201].
• AIM workshop Lipschitz metric on Teichmueller space, Problem 4.2, https://aimath.org/pastworkshops/lipschitzteichproblems.pdf. No proposer, year, remarks, or original literature reference is supplied. The extracted question is "Can we defining Thurston metric on $T_{qc}(\Gamma_0)$?"
• Supplied research attempt 1, registry updated 14 August 2026: partial result, open-at-run label, low novelty confidence, machine-generated and not peer reviewed. It supplies the non-domination equivalence and $D_c$ construction, and invokes standard quasiconformal length distortion and length-spectrum separation without complete authoritative citations.
• The verification field contains only "The exact canonical record is:" and supplies no additional hypotheses on $\Gamma_0$.