Symplectic duality and a complex-linearity obstruction for the Thurston metric

Definitions and mathematical statement

In the aid's setting, let $\mathcal T(S)$ be a positive-dimensional finite-type Teichmüller space of complete finite-area hyperbolic structures on an orientable punctured surface of negative Euler characteristic. Its Thurston metric and infinitesimal gauge are \[ d_{\mathrm{Th}}(x,y)=\log\sup_{\lambda\ne0} \frac{\ell_y(\lambda)}{\ell_x(\lambda)},\qquad F_x(v)=\sup_{\lambda\ne0}d_x\log\ell_\lambda(v), \] where $\lambda$ ranges over nonzero measured laminations. The canonical Problem 1.5, attributed to D. Dumas, asks in what sense this asymmetric metric is compatible with the complex structure of Teichmüller space, and asks the same for the symplectic structure. "Compatible" is not given a unique definition, and the finite-type hypotheses are supplied by the aid rather than the one-sentence question.

Let $g_{\mathrm{WP}},J,\omega_{\mathrm{WP}}$ be the Weil--Petersson metric, canonical complex structure and symplectic form. Use the normalization in which an earthquake vector $e_x(\lambda)$ satisfies \[ \Phi_x(e_x(\lambda))=d_x\ell_\lambda, \qquad\Phi_x(u)=\omega_{\mathrm{WP}}(\,\cdot\,,u). \] Then the supplied aid reports \[ \Phi_x\!\left(\frac{e_x(\lambda)}{\ell_x(\lambda)}\right) =d_x\log\ell_\lambda, \qquad F_x(v)=\max_{[\lambda]\in\mathbb P\mathcal{ML}(S)} \omega_{\mathrm{WP}}\!\left(v, \frac{e_x(\lambda)}{\ell_x(\lambda)}\right). \] Thus the gauge is a symplectic support function of the normalized earthquake vectors. The aid further states that $\Phi_x$ is a linear isometry from the earthquake tangent gauge to the dual Thurston cotangent gauge. The precise earthquake-gauge definition and asymmetric polar convention are not supplied and must be matched to this sign normalization; no identification with the Weil--Petersson Riemannian norm is intended.

With the Kähler convention $\omega_{\mathrm{WP}}(v,u)=g_{\mathrm{WP}}(Jv,u)$, the reported rotation identity is \[ J\!\left(\frac{e_x(\lambda)}{\ell_x(\lambda)}\right) =-\operatorname{grad}_{\mathrm{WP}}\log\ell_\lambda. \] Changes in earthquake or symplectic normalization may change signs or constants, so these must be checked together.

For a strong complex-compatibility interpretation, the aid invokes nowhere symmetry: at every such $x$, there exists $v$ with $F_x(v)\ne F_x(-v)$. It concludes that there is no real-linear $A:T_x\mathcal T(S)\to T_x\mathcal T(S)$ with \[ A^2=-I,\qquad F_x(Av)=F_x(v)\text{ for all }v. \] Indeed these two identities would force $F_x(-v)=F_x(A^2v)=F_x(v)$ for all $v$. In particular $J$ is not a gauge isometry. This rules out that specific infinitesimal isometric compatibility, not every possible relation with complex geometry. The aid also reports infinitesimal rigidity recovering the underlying hyperbolic/conformal surface up to isometry, without asserting recovery of its marking.

The extracted record includes a 21 April 2014 date, a WEIXU SU page header and the next section's statement that any two Teichmüller points can be joined by a geodesic concatenation of stretch segments. The supplied verification identifies this as page-boundary spillover; it is retained as neighboring context, not added as another clause of Dumas's problem.

Short English statement

The aid reports a symplectic duality through earthquake vectors, alongside an obstruction to making any linear complex structure an isometry of the asymmetric gauge. These address particular meanings of compatibility, while the original question is broader. The page-boundary geodesic text is kept as neighboring context.

Sources

• Local UnsolvedMath registry, permanent ID 20003090, source code catalogue entry; aim-topology-notes.json, notes[177].
• AIM, Lipschitz metric on Teichmueller space, Problem 1.5, D. Dumas, dated 21 April 2014 in the supplied extraction: https://aimath.org/pastworkshops/lipschitzteichproblems.pdf. The supplied verification reports inspecting page 1; this draft does not independently repeat it. The WEIXU SU header and following geodesic section are spillover.
• Earthquake--Weil--Petersson duality, nowhere asymmetry and infinitesimal Thurston rigidity are invoked in the supplied aid without full bibliographic entries or theorem-level hypotheses.
• Machine-generated research attempt 1: partial result, partially solved, rigor labelled "proved," low novelty confidence. Not peer reviewed; independently verify before citation. Registry updated 14 August 2026.