Finsler Realizations of the Bers Compactification

Definitions and mathematical statement

Let $S$ denote the surface intended in the source, and let $T(S)$ be its Teichmüller space of marked conformal structures, with markings taken up to the usual conformal equivalence and homotopy relation. The supplied record does not specify the topological type of $S$, its punctures or boundary, or the conventions for markings at boundary components. These parameters remain unresolved; no universal quantifier over surfaces is asserted here.

Fix the auxiliary data for a Bers embedding, including a reference conformal structure $X$, and write \[ b_X:T(S)\longrightarrow V_X, \qquad K_X=\overline{b_X(T(S))}, \] where $V_X$ is the space of holomorphic quadratic differentials used in that embedding and the closure is taken in its usual topology. The compactification $K_X$, together with $b_X$, is the corresponding Bers compactification. The reference structure and embedding conventions are not specified in the supplied record.

A Finsler structure $F$ assigns a tangent norm $F_p$ to each $p\in T(S)$, with the regularity required by the intended Finsler convention. Its induced path distance is \[ d_F(p,q)=\inf_{\gamma}\int_0^1 F_{\gamma(t)}\left(\dot\gamma(t)\right)\,{\,\mathrm{d}} t, \] where the infimum is over piecewise continuously differentiable paths $\gamma:[0,1]\to T(S)$ from $p$ to $q$. The source does not specify regularity, completeness, or whether reversibility $F_p(v)=F_p(-v)$ is required. If asymmetric Finsler structures are allowed, the tangent norms in this description are replaced by the corresponding possibly asymmetric Minkowski functionals.

For a chosen base point $o\in T(S)$, use the following horofunction convention: \[ h_y(z)=d_F(z,y)-d_F(o,y),\qquad \iota_F(y)=h_y,\qquad \mathcal H_F=\overline{\iota_F(T(S))}\subseteq C(T(S),{\mathbb{R}}), \] with closure in the topology of uniform convergence on compact subsets. For an asymmetric distance, using the opposite argument order can give a different construction; the supplied record does not select a convention.

Question. For the surface and Bers embedding intended by the source, does there exist a Finsler metric $F$ whose horofunction compactification is equivalent to the Bers compactification? With the conventions above, the requested equivalence means that $\mathcal H_F$ is a compactification of $T(S)$ and that there exists a homeomorphism \[ \Phi:\mathcal H_F\longrightarrow K_X \quad\text{such that}\quad \Phi\left(\iota_F(y)\right)=b_X(y) \quad\text{for every }y\in T(S). \] This interprets equality of compactifications as a homeomorphism extending the identification of their embedded interiors. The unspecified surface, Finsler conventions, auxiliary Bers data, and intended meaning of equivalence require confirmation before this becomes a fully determined formulation.

Short English statement

The question asks whether a Finsler metric on the intended Teichmüller space can recover the Bers compactification through its normalized distance functions. The formulation above interprets this as an equivalence of compactifications extending the identity on Teichmüller space. The surface, admissible Finsler metrics, and some compactification conventions remain unspecified in the supplied record.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001636$, source problem code catalogue entry, title Finsler Realizations of the Bers Compactification, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2010.
• Supplied report citation: Teichmüller Theory (2011), Oberwolfach Reports, pp. 3085--3157, DOI 10.4171/owr/2010/53. Original Oberwolfach problem report: https://doi.org/10.4171/owr/2010/53. The report title year and the recorded proposed year are retained as supplied.
• Supplied literature source, labeled "horofunction compactification of Teichmuller metric": https://arxiv.org/abs/1012.0409. No author names or fuller bibliographic details are supplied for this reference.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open, identifies the Teichmüller metric horofunction compactification with the Gardiner--Masur compactification rather than the Bers compactification, and describes the requested Finsler realization as unresolved. This is supplied dated curation, not an independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that its mathematical content was checked against the official Oberwolfach report, with notation, context, and question form normalized without changing the source-level claim. That verification statement is supplied provenance; no independent source consultation is claimed here.