Hitchin Curves and Weil–Petersson Boundary Dynamics

Definitions and mathematical statement

The supplied record poses two related research tasks, but does not specify enough data to determine a unique mathematical formulation.

A $C^1$ curve is a continuously differentiable map $\gamma:I\to M$, where $I\subseteq{\mathbb{R}}$ is an interval and $M$ is a differentiable manifold. A finite-dimensional family of such curves can be denoted by $(\gamma_a)_{a\in A}$, with a finite-dimensional parameter space $A$. The source does not specify $I$, $M$, the structure of $A$, or the dependence of $\gamma_a$ on $a$.

1. Hitchin curves. Find an ordinary differential equation describing at least some finite-dimensional families of $C^1$ curves arising from Hitchin representations. Here the collection of admissible curves remains unspecified: the source does not identify the surface or surface group, the target Lie group, or the construction associating curves to representations. It also does not specify the order of the equation, its dependent variables, its parametrization conventions, or the precise meaning of "describing" a family. These choices are necessary before an equation and its relation to the curves can be stated precisely. If derivatives beyond first order are required, additional regularity must also be specified.

2. Universal Weil--Petersson boundary dynamics. For universal Weil--Petersson Teichmüller space, determine the limit points in the Thurston boundary of Weil--Petersson geodesics, or give examples analogous to earthquake paths.

   For clarity, if a model $\mathcal T_{\mathrm{WP}}$, an ambient topological space $X$ containing it, and a specified Thurston boundary $D\subseteq X\setminus\mathcal T_{\mathrm{WP}}$ have been fixed, the boundary limit points of a geodesic $\eta:J\to\mathcal T_{\mathrm{WP}}$ can be expressed as \[ \operatorname{Acc}_{D}(\eta) = D\cap\overline{\eta(J)}^{\,X}. \] This is conditional notation, not an assertion that the source specifies such an $X$ or $D$. If limits along a particular end of $J$ are intended, that end and the corresponding limiting convention must additionally be specified. The source supplies neither the model and topology of the boundary nor the class and time domains of geodesics to be considered. It also gives no criterion for what makes a path analogous to an earthquake path. The disjunction between determining geodesic boundary limits and constructing earthquake-like examples is retained.

These tasks preserve the supplied formulation; the missing choices above remain unresolved and require human mathematical review.

Short English statement

Find an ordinary differential equation describing some finite-dimensional families of $C^1$ curves associated with Hitchin representations. Also, in universal Weil--Petersson Teichmüller space, determine boundary limits of Weil--Petersson geodesics or construct examples analogous to earthquake paths. The representations, curves, boundary topology, geodesic domains, and intended analogy are insufficiently specified in the supplied record.

Sources

• Local UnsolvedMath registry, supplied source record with permanent ID $30003958$, exact source problem code catalogue entry, title "Hitchin Curves and Weil--Petersson Boundary Dynamics," in the Oberwolfach Reports Open Problems collection.
• Supplied report citation: New Trends in Teichmüller Theory and Mapping Class Groups (2019), Oberwolfach Reports, pp. 2475--2534, DOI 10.4171/owr/2018/40. Supplied source URL: https://doi.org/10.4171/owr/2018/40. The registry records proposal year 2018; its report citation gives 2019. No individual proposer is supplied.
• Supplied literature reference, labelled "Oberwolfach Report 40, Burger problem": https://doi.org/10.4171/owr/2018/40. The label is retained as supplied attribution; it does not establish an individual proposer.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and reports that Hitchin and related higher Teichmüller literature describes the representation families, but that no ordinary differential equation answering the posed problem was found. 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 states that notation, typography, and omitted definitions were repaired while preserving mathematical content. Those are supplied editorial claims; this statement-only draft does not independently verify them or establish mathematical correctness.