Mean-Curvature Blowup Limits of Wings and Pitchfork Translators

Definitions and mathematical statement

For a smooth hypersurface flow $M_t\subset{\mathbb{R}}^{n+1}$, mean curvature flow means that its normal velocity equals its mean curvature vector: $(\partial_t X)^\perp=\mathbf H$. A translating soliton is a hypersurface $\Sigma$ with a nonzero constant vector $V$ such that $\Sigma_t=\Sigma+tV$ is a mean curvature flow; equivalently, $\mathbf H_\Sigma=V^\perp$. For a smooth hypersurface, its Gaussian entropy is \[ \lambda(\Sigma)=\sup_{x_0\in{\mathbb{R}}^{n+1},\,r>0}(4\pi r^2)^{-n/2}\int_\Sigma\exp\!\left(-\frac{|x-x_0|^2}{4r^2}\right)\,{\,\mathrm{d}}\mathcal H^n(x). \] Finite entropy means $\lambda(\Sigma)<\infty$. These definitions use a general dimension $n$; the supplied record does not explicitly specify the dimension of the requested flows.

Source question. Do the finite-entropy $\Delta$-wings and pitchfork translators of mean curvature flow arise as blow-up limits of mean curvature flows? The supplied dated assessment specifically identifies occurrence of pitchfork translators as blow-up limits of closed mean curvature flows as unresolved. Here closed means compact without boundary. This restriction is absent from the supplied question itself and must be retained as an unresolved qualification.

For an individual translator $\Sigma$, a possible precise realization condition would be the existence of a mean curvature flow $(M_t)$, points $x_j$, times $t_j$, and scales $a_j\to\infty$ such that the parabolically rescaled flows \[ \widetilde M^{(j)}_s=a_j\left(M_{t_j+a_j^{-2}s}-x_j\right) \] converge to $\Sigma+sV$ on a specified limiting time interval. The original flows must be defined at the displayed times. This is a formulation template, not a choice of the intended convergence: the source does not specify smooth or weak convergence, multiplicity, allowed centers and times, or the required relation to a singularity.

The supplied record names $\Delta$-wings and pitchfork translators without giving their geometric definitions or parameter ranges. Consequently, the intended families must be identified from the cited sources before this question becomes a fully specified existence statement. It is also unspecified whether every member of each family must occur, whether a separate originating flow may be used for each member, and whether finite entropy qualifies both families. The broader question about both families and the narrower dated assessment about pitchfork translators are preserved without identifying their scopes.

Short English statement

The question asks whether the specified $\Delta$-wing and pitchfork translating solutions can occur as limits obtained by magnifying mean curvature flows. The supplied assessment emphasizes flows of closed hypersurfaces for pitchfork translators. The intended families, quantifiers, and notion of blow-up convergence require clarification before a fully precise statement can be fixed.

Sources

• Local registry provenance: supplied UnsolvedMath record, permanent ID $30004190$, source code catalogue entry, in the Oberwolfach Reports Open Problems collection. Project registry locations are data/unsolvedmath/problems.json and catalog/problems.json; their contents were not independently inspected for this draft.
• Supplied source citation: Partial Differential Equations (2020), Oberwolfach Reports, pp. 2033--2097, DOI 10.4171/owr/2019/34. Original Oberwolfach report: https://doi.org/10.4171/owr/2019/34. The record describes these reports as documenting mathematical workshops and their open-problem discussions. It gives proposed year 2019 and supplies no individual proposer.
• Supplied literature source: Scherk-like Translators for Mean Curvature Flow, https://arxiv.org/abs/1903.04617.
• Supplied literature source: Uniqueness of Semigraphical Translators, https://arxiv.org/abs/2310.06980.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open, states that the translators and their uniqueness are established, and describes occurrence of pitchfork translators as blowup limits of closed mean curvature flows as unresolved. This is supplied dated curation, not an independently verified literature assessment. The source explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says its mathematical content was preserved while notation, typography, duplicate clauses, or omitted definitions were repaired for self-contained publication. This records a supplied editorial assessment, not mathematical verification of this draft.