Ancient Ovals as Post-Singularity Limit Flows

Definitions and mathematical statement

A smooth hypersurface mean curvature flow in Euclidean space is a family of immersions $X(\cdot,t):M^n\to{\mathbb{R}}^{n+1}$ whose normal velocity equals its mean curvature vector: $(\partial_tX)^\perp=\vec H$. An ancient solution is a solution defined for all times $s\in(-\infty,S)$, for some finite $S$. The term ancient oval ordinarily refers to a compact, convex ancient solution that is not a self-similarly shrinking round sphere. The supplied record does not specify the dimension or the additional geometric hypotheses intended by this term, so this description does not resolve the intended ancient-oval class.

Let $T_1$ denote the first singular time of an initially smooth flow. Discussing a singularity at a time $t_*>T_1$ requires a continuation of the flow through its first singular time; no continuation framework is specified in the source record.

A usual form of parabolic rescaling is \[ \mathcal M^{(j)}_s =\lambda_j\left(M_{t_j+\lambda_j^{-2}s}-x_j\right), \qquad \lambda_j\to\infty, \qquad (x_j,t_j)\to(x_*,t_*). \] A limit of such rescalings, in an appropriate convergence topology, is one possible meaning of a limit flow at $(x_*,t_*)$. Fixing every center to $(x_*,t_*)$ gives the more specific tangent-flow construction. The record does not specify the permitted centers, scales, convergence, or whether this more specific construction is intended.

The supplied question is: Can ancient ovals arise as limit flows at singularities of a mean curvature flow after the first singular time?

Its existential target is a continued mean curvature flow, a singular spacetime point $(x_*,t_*)$ with $t_*>T_1$, and a limiting ancient oval arising from rescalings at that singularity. The ambient dimension, admissible initial flows, continuation, ancient-oval class, and meaning of limit flow remain unspecified. Consequently, this target is a statement schema requiring those choices, rather than a uniquely determined quantified proposition.

Short English statement

The question asks whether an ancient oval can occur as a blow-up limit at a singularity strictly later than the first singular time of a mean curvature flow. A precise formulation requires the intended dimension, class of flows, continuation through singularities, and definition of blow-up limit. The supplied record labels the problem open; that status is supplied dated curation, not independently verified here.

Sources

• Local UnsolvedMath registry, supplied complete record for permanent ID $30004196$, source problem code catalogue entry, title Ancient Ovals as Post-Singularity Limit Flows, in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2019 and supplies no individual proposer attribution.
• Supplied report citation: Partial Differential Equations (2020), Oberwolfach Reports, pp. 2033--2097, DOI 10.4171/owr/2019/34. https://doi.org/10.4171/owr/2019/34. The supplied background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source, labeled "original Oberwolfach report": https://publications.mfo.de/bitstream/handle/mfo/3782/OWR_2019_34.pdf?isAllowed=y&sequence=1.
• Supplied literature source: Ancient Low Entropy Flows, Mean Convex Neighborhoods, and Uniqueness. https://arxiv.org/abs/1810.08467.
• Supplied literature source: Rigidity of Ancient Ovals in Higher Dimensional Mean Curvature Flow. https://arxiv.org/abs/2504.09741.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and reports that mean-convex-neighborhood and ancient-flow classification results have advanced, while occurrence of ancient ovals at subsequent singularities remains explicitly unresolved. This is supplied dated curation derived from the OWR record, not an independently verified assessment. The supplied background requests independent verification of time-sensitive status and cited claims.