Mean curvature flows with prescribed singular sets

Raphael Tsiamis

arxiv: 2604.14139 · v1 · submitted 2026-04-15 · 🧮 math.DG · math.AP

Mean curvature flows with prescribed singular sets

Raphael Tsiamis This is my paper

Pith reviewed 2026-05-10 11:42 UTC · model grok-4.3

classification 🧮 math.DG math.AP MSC 53C44

keywords mean curvature flowancient solutionssingular setsmean-convex hypersurfacesmetric perturbationprescribed singularitiesgeometric flowshypersurface evolution

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The pith

Mean-convex ancient mean curvature flows can be constructed so their first singularities occur exactly on any prescribed closed set by a small metric change.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows how to build, for any closed set K in R to the n and any m at least 2, an ancient mean-convex solution to mean curvature flow of hypersurfaces in R to the m plus n. The ambient space uses a smooth Riemannian metric that stays arbitrarily close to flat Euclidean space. The solution first becomes singular precisely along the product of K with the origin in the extra dimensions. This construction matters because it demonstrates that the locations where these flows develop singularities can be controlled with high precision through minor adjustments to the geometry of the surrounding space.

Core claim

For every closed set K subset of R^n and every m greater than or equal to 2, there exists a mean-convex ancient solution to the mean curvature flow of hypersurfaces in R^{m+n} with respect to a smooth Riemannian metric that is arbitrarily C^infty-close to the Euclidean metric, and the first-time singular set of this solution is exactly K times the zero vector in the additional m dimensions.

What carries the argument

The mean-convex ancient solution to mean curvature flow whose first singular set is forced to equal exactly the prescribed product K times zero via a C^infty-small perturbation of the Euclidean metric on the ambient space.

If this is right

Any closed set in R^n, no matter how irregular, can be realized as the precise first singular set of some ancient mean-convex flow in sufficiently high ambient dimension.
Small changes to the background metric suffice to dictate singularity locations without destroying the mean-convexity or ancient character of the solution.
The construction applies uniformly across all closed sets and works when the hypersurface has codimension at least two.
Ancient solutions exist whose singularity sets can be chosen independently of the topology or dimension of the ambient Euclidean space.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Singularity formation in mean curvature flow appears more flexible than previously thought, since the ambient geometry can be tuned to place singularities wherever desired.
The same perturbation technique might be adapted to prescribe singularities in other parabolic geometric flows such as Ricci flow or mean curvature flow in other ambient spaces.
Numerical simulations for concrete choices of K, such as Cantor sets, could test whether the constructed flows remain stable under further small perturbations.
This flexibility raises the question of whether generic initial data in perturbed metrics naturally produce complicated singular sets that can be classified by their Hausdorff dimension.

Load-bearing premise

A smooth metric perturbation arbitrarily close to Euclidean exists that forces the first singular set of the ancient mean-convex flow to be exactly the prescribed K times zero for arbitrary closed K.

What would settle it

Exhibiting one specific closed set K together with a proof that no ancient mean-convex solution exists with first singular set exactly K times zero no matter how the ambient metric is perturbed while remaining smooth and close to Euclidean.

read the original abstract

For every closed set $K \subset \mathbb{R}^n$ and every $m \geq 2$, we construct a mean-convex ancient solution to mean curvature flow of hypersurfaces in $\mathbb{R}^{m+n}$, with respect to a smooth Riemannian metric arbitrarily $C^\infty$-close to the Euclidean metric, whose first-time singular set is exactly $K \times \{0\}$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper constructs ancient mean-convex MCF solutions whose first singular set is any prescribed closed K by a small metric perturbation from Euclidean.

read the letter

The main takeaway is that the authors construct ancient mean-convex solutions to mean curvature flow whose first singular set is an arbitrary closed set K, by using a Riemannian metric that is arbitrarily close to the Euclidean one. This holds in R^{m+n} for m at least 2, with the singular time normalized to zero on K times the origin in the extra dimensions. Previous work on prescribed singularities in MCF was limited to points, curves, or other low-dimensional or smooth sets, so the arbitrary closed K is the step forward. It gives a way to realize complicated singularity patterns on demand while keeping the flow mean-convex and ancient. The metric perturbation approach looks reasonable on the surface because it lets you adjust the geometry locally without altering the overall topological or convexity setup too much. Higher codimension helps with room to maneuver the estimates. A soft spot is the precise control over the singular set for arbitrary K. Closed sets can be fractal or have empty interior, so ensuring singularities appear exactly on K and nowhere earlier or elsewhere requires tight bounds on curvature blow-up and distance to the set. Mean-convexity must survive the perturbation without creating new singularities or smoothing out parts of K. The abstract states the result but does not display the barrier arguments or parabolic estimates, so it is hard to see how they close the construction for the worst-case K. This paper is for people working on singularity formation and ancient solutions in geometric flows. A reader who wants concrete examples of complicated singularities would get value from it if the details check out. It deserves serious refereeing because the claim is specific and the method is standard enough for experts to verify the estimates.

Referee Report

0 major / 1 minor

Summary. The manuscript claims that for every closed set K ⊂ R^n and every integer m ≥ 2, there exists a mean-convex ancient solution to mean curvature flow of hypersurfaces in R^{m+n} equipped with a smooth Riemannian metric that is arbitrarily C^∞-close to the Euclidean metric, such that the singular set at the first singular time is exactly K × {0}.

Significance. If the construction and verification hold, the result would establish substantial flexibility in the singularity formation of mean-convex ancient MCF under small ambient metric perturbations, allowing arbitrary closed singular sets. This could influence the study of singularity models and the classification of ancient solutions in geometric flows.

minor comments (1)

The abstract states the main existence result clearly, but the manuscript should include an explicit outline of the metric perturbation construction and the estimates ensuring mean-convexity is preserved while forcing the exact singular set K × {0}.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their report. The summary accurately reflects the main theorem. No specific major comments were provided, and the recommendation is listed as uncertain. We interpret this as possible concern over the technical details of the construction or the verification that the singular set is precisely K × {0}. We are prepared to supply additional explanations or minor clarifications in a revision.

Circularity Check

0 steps flagged

No significant circularity in derivation

full rationale

The paper presents an existence construction for mean-convex ancient MCF solutions in a perturbed metric with exactly prescribed first singular set K × {0}. No equations, parameter fits, self-citations, or ansatzes are visible that reduce the claimed result to the target singular set by definition or construction. The central claim is a perturbation-based existence theorem whose steps remain independent of the output singular set.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete; no free parameters, new entities, or non-standard axioms are mentioned.

axioms (2)

standard math Standard existence, uniqueness, and regularity theory for mean curvature flow of hypersurfaces holds in Riemannian manifolds close to Euclidean space.
The construction presupposes the well-posedness of MCF under small metric perturbations.
domain assumption Ancient mean-convex solutions can be constructed or glued so that their first singularity coincides with a prescribed closed set after metric adjustment.
This is the core modeling assumption needed to achieve exact control over the singular set.

pith-pipeline@v0.9.0 · 5340 in / 1391 out tokens · 28293 ms · 2026-05-10T11:42:56.054565+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

[1]

Huang and X

[HZ25] Y. Huang and X. Zhao. On the rate of convergence of cylindrical singularity in mean curvature flow.Preprint arXiv:2510.23499,

work page arXiv
[2]

[Whi02] B. White. Evolution of curves and surfaces by mean curvature. In : Proceedings of the International Congress of Mathematicians (Beijing, 2002)

work page 2002

[1] [1]

Huang and X

[HZ25] Y. Huang and X. Zhao. On the rate of convergence of cylindrical singularity in mean curvature flow.Preprint arXiv:2510.23499,

work page arXiv

[2] [2]

[Whi02] B. White. Evolution of curves and surfaces by mean curvature. In : Proceedings of the International Congress of Mathematicians (Beijing, 2002)

work page 2002