Discretization of Elastic Curves in Shape Spaces

Definitions and mathematical statement

An infinite-dimensional shape space is the space of admissible geometric curves contemplated by the source; the record does not specify its elements, regularity, topology, or whether curves are identified under reparametrizations or other transformations. The term continuous elastic-bending variational problem refers to the continuum variational problem for these curves, but its functional, admissible variations, constraints, and boundary conditions are not supplied.

Research problem. Construct a discretization of elastic curves in the intended infinite-dimensional shape space that is compatible with the associated continuous elastic-bending variational problem.

For schematic notation only, let $\mathcal S$ denote the intended continuous shape space and $E$ its intended variational functional. The requested construction would supply discrete curve spaces $\mathcal S_h$ and corresponding discrete variational problems, with $h$ denoting a discretization parameter tending to zero. These symbols do not specify a model or impose additional hypotheses. In particular, the source does not define what compatibility between the discrete and continuous problems must mean: it supplies no required approximation topology, convergence statement, or criterion involving energies, minimizers, stationary curves, or variational equations. Consequently, a precise quantified existence or convergence assertion cannot be extracted without further specifications. The record also does not determine whether the intended construction concerns a particular shape space, a specified class, or all elastic shape spaces.

Short English statement

Develop discrete models of elastic curves that faithfully represent their continuous elastic-bending variational problem. The supplied record leaves the curve class, shape space, energy, boundary conditions, and meaning of compatibility unspecified, so it determines a research question rather than a fully specified mathematical assertion.

Sources

• Local UnsolvedMath registry, supplied complete source record, permanent ID $30003039$, source problem code catalogue entry, titled Discretization of Elastic Curves in Shape Spaces; collection: Oberwolfach Reports Open Problems. The supplied proposed year is 2016; no individual proposer is identified.
• Supplied report provenance: Mathematical Imaging and Surface Processing (2016), Oberwolfach Reports, pp. 155--214, DOI 10.4171/owr/2016/4. https://doi.org/10.4171/owr/2016/4. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions; no problem-specific page or passage is supplied.
• Supplied literature reference: Rumpf and Wirth, Variational time discretization of geodesic calculus. https://arxiv.org/abs/1210.2097.
• Supplied literature reference: A discrete framework for matching manifold-valued curves. https://arxiv.org/abs/1703.05107.
• Supplied literature reference: Oberwolfach Report 4/2016. https://doi.org/10.4171/owr/2016/4.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem partially solved and states that variational time discretization, discrete curve models, and convergence results provide principled discretizations for important infinite-dimensional shape spaces, but do not provide a single general discretization for every elastic shape space. This is supplied dated curation derived from the OWR record, not an independently verified assessment. Its time-sensitive status and cited claims require independent verification.