Continuity of Thurston geodesic envelopes and diagonal collapse

Definitions and mathematical statement

Fix a finite-type orientable surface $S$ in the hyperbolic range and its finite-area Teichmüller space. With Thurston's asymmetric metric \[ d_{\rm Th}(X,Y)=\log\sup_{\gamma}\frac{\ell_\gamma(Y)}{\ell_\gamma(X)}, \] define the geodesic envelope as the union of all directed geodesic segments from $X$ to $Y$. Equivalently, since geodesics exist, it is the metric interval \[ \operatorname{Env}(X,Y)=\{Z:d_{\rm Th}(X,Z)+d_{\rm Th}(Z,Y)=d_{\rm Th}(X,Y)\}. \] The question attributed to K. Rafi asks whether this compact-set-valued map is continuous in both endpoints. Make the topology explicit by setting $d_s(X,Y)=\max\{d_{\rm Th}(X,Y),d_{\rm Th}(Y,X)\}$ and using the Hausdorff metric of $d_s$ on compact sets. In zero-dimensional Teichmüller space the assertion is immediate.

The supplied aid reports Hausdorff continuity for the once-punctured torus and for closed orientable surfaces of genus at least two. It says the literature checked did not cover all higher-complexity punctured surfaces in the original finite-type scope; this is a limitation of the supplied check, not a new assertion that no theorem exists. For the closed case it identifies Pan--Wolf's headline Theorem 1.1 as unrestricted and detailed Theorem 5.1 as stated off the diagonal.

The aid supplies an abstract diagonal argument. Let $(M,d)$ be a directed geodesic asymmetric metric space, with continuous distance, compact closed forward balls in the symmetrized topology, and compact-set reversibility: for every compact $K$ there is $C_K\ge1$ such that \[ d(b,a)\le C_Kd(a,b)\qquad(a,b\in K). \] Let $I_d(x,y)=\{z:d(x,z)+d(z,y)=d(x,y)\}$. Its intervals are compact and upper semicontinuous in endpoints: if $x_n\to x,y_n\to y$, no points of $I_d(x_n,y_n)$ can remain a positive symmetrized distance from $I_d(x,y)$. For $x_n,y_n\to x$, a single suitable compact $K$ contains the relevant intervals eventually, and \[ d_H^s\left(I_d(x_n,y_n),\{x\}\right) \le d_s(x,x_n)+C_Kd(x_n,y_n)\longrightarrow0. \] Thus the aid obtains diagonal continuity for Thurston envelopes under these standard finite-type metric hypotheses, quantitatively separating it from the off-diagonal issue. Upper semicontinuity alone does not imply full continuity: lower semicontinuity requires each point of the limiting interval to be approximated by points of nearby intervals. For punctured cases not covered by the supplied references, that off-diagonal lower-semicontinuity step remains the unresolved target in the report.

Short English statement

Ask whether all Thurston geodesics between two endpoints form a Hausdorff-continuous envelope. The supplied aid reports continuity in stated surface classes and proves diagonal collapse under properness and local reversibility, while distinguishing this from off-diagonal lower semicontinuity in unverified punctured cases.

Sources

• K. Rafi, Problem 2.3, AIM workshop Lipschitz metric on Teichmueller space, https://aimath.org/pastworkshops/lipschitzteichproblems.pdf.
• The research aid cites Pan--Wolf, headline Theorem 1.1 and detailed Theorem 5.1, for closed surfaces, and refers to known once-punctured-torus continuity without a full bibliographic entry for either source.
• Local UnsolvedMath registry, permanent ID 20003093, exact code catalogue entry; canonical location aim-topology-notes.json, notes[180], dated 14 August 2026; statement labeled exact.
• Research attempt 1: partial result, problem partially solved at the run, rigor label proved, novelty confidence low; registry status partially solved. The abstract lemma and literature-scope account are machine-generated research aids, not peer reviewed; independent verification is required before citation.