Non-uniqueness of Brakke flows starting from minimal surfaces with singularities

Kotaro Motegi

REVIEW 1 major objections 5 minor 13 references

Non-uniqueness of Brakke flows starting from minimal surfaces with singularities

T0 review · 1 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper proves that a stationary minimal hypersurface with a singular point of density greater than 1 and sufficiently small scale-invariant $L^2$ distance to a plane admits a genuinely time-dependent Brakke flow.

desk verdict A substantial generalization of Stuvard–Tonegawa with a fresh integrality argument, but the proof of the key slab estimate (4.2) is asserted rather than derived. read the letter →

arxiv 2608.13531 v1 pith:B4IMMN3J submitted 2026-08-13 math.AP math.DG

Kotaro Motegi This is my paper · ORCID

classification math.APmath.DG MSC 49Q1553E1035K55

keywords Brakkeflowmeancurvaturedynamicalinstabilitystationaryvarifoldnon-uniquenessminimalsurfacesingularitytangentcone

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper establishes that a stationary minimal hypersurface $\Gamma_0$ with a singular point where its density exceeds 1 and its scale-invariant $L^2$ distance to some plane has sufficiently small limsup is dynamically unstable in Brakke's weak sense. Dynamical instability means there exists a non-trivial Brakke flow—one whose mass is strictly less than the initial mass for every $t>0$—starting from $\Gamma_0$. Since $\Gamma_0$ is stationary, the constant flow is already a Brakke flow, so this gives non-uniqueness of weak mean curvature flows from the same initial datum. The proof removes a logarithmic-decay assumption on tangent cones used in earlier work and does not require the tangent cone at the singularity to be unique.

What carries the argument

The argument runs through a hole-nucleation construction: a Lipschitz map $g_\varepsilon$ modifies $\Gamma_0$ inside a tiny ball, producing a new surface $\Gamma^\varepsilon_0$ that agrees with $\Gamma_0$ outside $U_{2\varepsilon}$ and has a hole at the origin. Brakke's expanding holes lemma, with a Gaussian test function $\Phi_\lambda$, bounds the weighted area of the flow at later times by its initial value plus an error term controlled by the scale-invariant $L^2$ excess from $T$. Huisken's monotonicity formula supplies uniform-in-time density and excess estimates for the approximating flows. The decisive new ingredient is integrality: in a blow-up limit, a varifold supported on the plane $T$ with bounded energy must have constant integer density, so its density ratio is forced to be $0$ or $1$; this contradicts the level-crossing at $1+\delta$ that would occur if the first hitting time $T_\varepsilon$ collapsed to zero. The integrality of varifolds is what replaces the logarithmic-decay assumption of earlier work.

What would settle it

Check whether the slab estimate (4.2) holds for a concrete stationary integral cone with density $3/2$ that satisfies the $L^2$-excess condition (3.1), such as three half-planes meeting along a common axis at small angles: compute the Hausdorff distance from the plane $T$ over $U_{2\varepsilon}$. If for arbitrarily small $\varepsilon$ the cone is not contained in $\{|x_{n+1}|\le \varepsilon/20\}$ despite having small $L^2$ distance, then Lemma 4.1 cannot supply the uniform excess and density bounds (4.1) and (4.4), and the proof of Theorem 3.4 collapses.

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Extended reading notes

Core claim

The central claim is Theorem 3.4: for any $\Theta_0>1$ there is a constant $\mu_0=\mu_0(n,\Theta_0)\in(0,1)$ such that if $\Gamma_0$ is a closed countably $n$-rectifiable set whose associated multiplicity-one varifold is stationary, whose density at the origin belongs to $(1,\Theta_0]$, and which satisfies the small-$L^2$-excess condition $\limsup_{R\to 0} R^{-n-2}\int_{\Gamma_0\cap U_R} |T^\perp x|^2\,d\mathcal{H}^n(x)\le \mu_0^2$ for some $n$-plane $T$, then there exists a Brakke flow $\{V_t\}_{t\ge 0}$ with fixed boundary $\partial\Gamma_0$ such that $\lim_{t\downarrow 0}\|V_t\|=\|V_0\|=\mathcal{H}^n\lfloor \Gamma_0$ and $\|V_t\|(U)<\|V_0\|(U)$ for all $t>0$. The genuinely time-dependent flow is obtained as a limit of Brakke flows starting from modified surfaces with a small hole cut at the singular point, and the strict mass loss shows that the static flow is not the only weak evolution.

Load-bearing premise

The load-bearing premise is that a regularity estimate from Allard's theory holds for the singular point—namely that near the origin the entire surface lies in a slab of width proportional to the scale around the planar direction $T$—even though the density there exceeds 1, so Allard's classical regularity theorem does not directly apply.

Editorial extensions

If this is right

If the theorem is correct, uniqueness of Brakke flows fails for every stationary hypersurface satisfying Assumption 3.2: the static flow is never the only weak evolution from such an initial datum.
The logarithmic decay condition on tangent cones from previous work becomes unnecessary; only a small limsup of the $L^2$ excess at the singular point is needed.
The result applies to broad natural classes—stationary two-valued Lipschitz graphs, stable codimension-one integral varifolds, and area-minimizing mod-$p$ hypersurfaces—whenever existing tangent-cone uniqueness results place them in Assumption 3.2.
The constructed flow loses mass immediately: $\|V_t\|(U)<\|V_0\|(U)$ for every $t>0$, so the singularity does not merely delay motion but initiates it right away.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper: the proof's waiting-time estimate $T_\varepsilon\ge \delta_0\varepsilon^2$ and the blow-up contradiction suggest a quantitative lower bound on the rate of mass loss near $t=0$; one could try to extract an explicit rate from the constants in Lemmas 7.1 and 7.3.
Beyond the paper: the level-crossing argument uses integrality to rule out density ratios strictly between $0$ and $1$; a similar argument with thresholds $kc$ for integer $k$ might probe singularities of higher density where the limit density could jump across integer levels.
Beyond the paper: the slab estimate (4.2) is the only step that imports pointwise closeness from $L^2$ flatness; an $L^2$-only hole construction would remove the most delicate assumption and likely cover unions of half-planes at small angles without invoking a classical regularity theorem.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Let me tell you what I think of Motegi's paper.

The new thing here is real. The paper takes Stuvard–Tonegawa's hole-nucleation strategy for dynamical instability and pushes it past two restrictions that were clearly artificial: the logarithmic convergence rate to a flat tangent cone, and tangent-cone uniqueness. The key novelty is the integrality argument in Lemmas 7.2–7.3. The idea is to use the fact that a stationary integral varifold with small L2 distance to a plane has density close to an integer, so it cannot cross the threshold 1+δ at arbitrarily small times. That is a genuinely nice piece of reasoning, and it is what removes the log-decay assumption. The paper is also well organized, and the estimates in Sections 5 and 6 are adapted carefully from the standard toolkit.

The main soft spot is the pointwise slab estimate (4.2) in Lemma 4.1. The proof just asserts that Allard's [All72, 7.5] gives Γ0 ∩ U_2ε ⊂ {|x_{n+1}| ≤ ε/20} under Assumption 3.2. The trouble is that Allard's height/regularity estimate normally requires the mass ratio in the ball to be close to 1, and here the density at the origin is >1. L2 closeness does not by itself give pointwise closeness; a small set of high points could still have small L2 excess. This might be fixable, e.g., via a monotonicity argument: a point at height h would carry mass at least c h^n, which would violate the L2 bound for μ small. But the author does not supply that argument, and (4.2) is load-bearing for the hole construction. As it stands, this is a genuine gap that a referee should close before acceptance.

Smaller issues: Lemma 7.1 states ||V^ε_{T_ε}||(Φ_ε(·,T_ε)) = αc, but if T_ε = ∞ the value at T_ε is undefined. Not a big deal, because the contradiction in Lemma 7.3 only runs when T_ε is finite, but the lemma as written overreaches. The abstract also omits Assumption 3.1 (the Caccioppoli partition and boundary conditions), so the advertised statement is slightly stronger than Theorem 3.4.

Overall, the proof is coherent once you grant (4.2), and the gap looks bridgeable rather than fatal. This is a significant paper for the GMT community and deserves a careful referee. I would send it to review and ask for a justification of (4.2), either a proof or a precise citation covering the density >1 case.

Referee Report

1 major / 5 minor

Summary. The paper proves that a stationary multiplicity-one varifold associated to a closed countably n-rectifiable set Γ0 is dynamically unstable (and hence has non-unique Brakke flows) provided that the density at a singular point exceeds 1 and that the scale-invariant L2 distance of Γ0 from some n-plane has sufficiently small limsup at the origin. The proof follows the framework of Stuvard and Tonegawa: one constructs modified initial surfaces with a hole, obtains Brakke flows starting from them, uses Brakke's expanding holes lemma and Huisken's monotonicity formula to get uniform density and L2-excess estimates, and then runs a blow-up argument. The new ingredient is an integrality-based density-gap lemma that prevents the density ratio from crossing a threshold near the initial time, removing the logarithmic decay condition from [ST25] and avoiding any uniqueness assumption on tangent cones.

Significance. If the gap described below is repaired, the result constitutes a substantial advance: it broadens the class of singularities known to be dynamically unstable, removes the logarithmic decay assumption of [ST25], and applies to several well-studied classes (two-valued Lipschitz graphs, stable codimension-one integral varifolds, area-minimizing mod p hypersurfaces). The integrality argument in Lemma 7.2 is elegant and the overall structure of the proof is transparent. The paper is generally well written and clearly exposes its reliance on external results. However, the present version contains a load-bearing unproved assertion in Lemma 4.1, namely the pointwise slab estimate (4.2), and this must be addressed before the theorem can be considered established.

major comments (1)

[Lemma 4.1, Eq. (4.2)] The pointwise slab bound Γ0∩U_{2ε}⊂{|x_{n+1}|≤ε/20} is asserted directly from [All72, 7.5(2),(6)] and Assumption 3.2. Allard's height estimate in that form is a regularity theorem that applies when the mass ratio is close to 1, whereas Assumption 3.2 permits the density at the origin to be any value in (1,Θ0]. The L2 excess condition (3.1) alone does not imply a pointwise or Hausdorff closeness statement of the type (4.2); for example, two unit-density planes crossing at angle θ have L2 excess comparable to sin^2 θ while their maximum height at radius ε is ε sin θ. The author needs to prove a quantitative bound of the form |x^⊥| ≤ C μ |x| near 0, or supply a precise citation that covers the density >1 case, because (4.2) is used to obtain the uniform estimates (4.1), (4.4), and property (5) of Lemma 4.1, on which all subsequent sections rely.

minor comments (5)

[Lemma 7.2, Eq. (7.6)] In the statement of Lemma 7.2, equation (7.6) is written with d∥V_t∥ but should be d∥V∥; the lemma concerns a single varifold V, not a flow.
[Definition 2.1(4)] In the displayed Brakke inequality (2.4), the integrand should be parenthesized as (−ϕ(x,t)h(V_t,x)+∇ϕ(x,t))·h(V_t,x)+∂_tϕ(x,t) for readability and to avoid ambiguity in the sign of the ∂_t term.
[Proposition 4.3] Proposition 4.3 is stated before the uniform mass estimates of Section 6 are available; the compactness argument is only cited from [ST25, Proposition 4.3]. The author should either move this proposition after Section 6 or explicitly indicate that the needed uniform mass bounds follow from Proposition 4.2 together with Lemma 4.1.
[Proof of Theorem 3.4, final paragraph] The approximation argument that produces a compactly supported test function ψ from 1−Φ_0(·,s_0) is only sketched. A short justification using monotone convergence with càdlàg cutoffs would make the argument complete.
[Lemma 7.3] When the rescaled test function Φ_{T_ε^{-1/2} ε_j} first appears, it has not been defined; the author should state explicitly that it is the function Φ_λ from Section 5 evaluated with λ = T_ε^{-1/2} ε_j.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the theorem is derived from independent external tools and a new integrality argument; no step reduces to its inputs by construction.

full rationale

The central claim (Theorem 3.4) is not an input to Assumption 3.2: the assumption only posits stationarity, density greater than 1, and the scale-invariant L2 closeness (3.1). The parameter mu0 is chosen sufficiently small in Lemma 7.3 after determining the constants E, Gamma, Lambda, and nu from n, Theta0, zeta, and alpha; it is not fitted to the target mass loss or to the final inequality. The hole construction in Lemma 4.1 imports the map g and the sets A and B from [ST25, Lemma 4.1], and the expanding-holes estimate in Lemma 5.1 uses Brakke's test-function together with [ST25, Lemma 5.3] and [KT14]/[Ton19] estimates; all of these are external published results used as tools, and none of them presupposes Theorem 3.4. The integrality contradiction in Lemma 7.2 is the new content: Allard compactness and integrality of the limit force the density to be an integer, ruling out the threshold value alpha c; this is not a restatement of the assumption. The only delicate point in the paper is the invocation of [All72, 7.5] to obtain the pointwise slab bound (4.2) at a point where the density exceeds 1; this is a potential correctness or regularity concern, but it is not circularity, since it is an external estimate applied to the initial varifold and not the conclusion being derived. Accordingly, the derivation chain is self-contained in the relevant sense and no prediction is equivalent by construction to its input.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The proof imports a large body of standard geometric measure theory results: Brakke flow existence and compactness, Allard's compactness and regularity, Brakke's perpendicularity and expanding holes lemma, Huisken's monotonicity formula and the heat-kernel estimates from [KT14]. The only free numerical parameter is the smallness constant mu0, which is existential, not fitted. No new entities are postulated. The most restrictive structural assumption is Assumption 3.1, which requires Gamma0 to be the reduced boundary of a Caccioppoli partition in a strictly convex C2 domain with fixed boundary; this is not reflected in the abstract.

free parameters (1)

mu0
Smallness threshold in Theorem 3.4 for the L2 flatness ratio (3.1). It is an existential constant produced by the proof (Lemma 7.3), not a number fitted to data. It depends only on n and Theta0.

assumptions (6)

standard math Existence and compactness theory for Brakke flows with fixed boundary ([ST21, Theorems 2.2-2.3], [Ilm94], [Ton19, Section 3.3])
Used in Proposition 4.2 to produce flows from the modified surfaces Gamma_epsilon_0 and in Proposition 4.3 to extract a limit flow. This is a standard external theory.
standard math Allard's compactness and height-estimate results, including [All72, 7.5 (2) and (6)]
Provides the slab estimate (4.2) in Lemma 4.1 and the compactness passage in Lemma 7.2. Its applicability at density greater than 1 is not demonstrated in the text.
standard math Brakke's perpendicularity theorem and expanding holes lemma ([Bra78, 5.8, 6.5])
Used in Lemma 5.1 to derive the weighted area growth estimate through the hole.
standard math Huisken's monotonicity formula and heat kernel estimates ([Sim83, 17.5], [KT14, Propositions 6.2 and 6.4])
Used in Section 6 to obtain uniform density ratio and L2 excess bounds, in particular Proposition 6.2.
domain assumption Assumption 3.1: Gamma0 is the union of boundaries of a Caccioppoli partition {E_{0,i}} with reduced-boundary coverage H^n(Gamma0 \ union d*E_{0,i}) = 0 in a strictly convex C2 domain with non-empty boundary
This global structural hypothesis is required for the existence theorem [ST21, Theorem 2.2] used to start the approximating flows. It is not implied by stationarity and is omitted from the abstract.
domain assumption Assumption 3.2(1): the multiplicity-one varifold V0 = var(Gamma0, 1) is stationary
Stationarity makes the static family a Brakke flow and provides the monotonicity formula used for density estimates.

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Cite this review

Pith. "Pith review of Non-uniqueness of Brakke flows starting from minimal surfaces with singularities." pith.science (2026). https://pith.science/paper/B4IMMN3J

@misc{pith2026260813531,
  author       = {Pith},
  title        = {Pith review of: Non-uniqueness of Brakke flows starting from minimal surfaces with singularities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/B4IMMN3J}},
  note         = {Machine review of arXiv:2608.13531}
}

abstract

We prove the existence of a genuinely time-dependent Brakke flow starting from $\Gamma_0 \subset \mathbb{R}^{n+1}$ whose associated multiplicity-one varifold is stationary, provided that, at some singular point, the scale-invariant $L^2$ distance of $\Gamma_0$ from an $n$-dimensional plane has sufficiently small limsup as the scale tends to zero. This yields the dynamical instability of $\Gamma_0$, a notion recently introduced by Stuvard and Tonegawa, and hence the non-uniqueness of Brakke flows starting from $\Gamma_0$. A notable feature of our result is that it holds without assuming the uniqueness of tangent cones at the singular point.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

William K. Allard. On the first variation of a varifold. Ann. of Math. (2) , 95:417--491, 1972

1972

[2] [2]

Becker-Kahn

Spencer T. Becker-Kahn. Transverse singularities of minimal two-valued graphs in arbitrary codimension. J. Differential Geom. , 107(2):241--325, 2017

work page 2017

[3] [3]

On the nature of stationary integral varifolds near multiplicity 2 planes, 2025

Spencer Becker-Kahn, Paul Minter, and Neshan Wickramasekera. On the nature of stationary integral varifolds near multiplicity 2 planes, 2025

2025

[4] [4]

Kenneth A. Brakke. The motion of a surface by its mean curvature , volume 20 of Mathematical Notes . Princeton University Press, Princeton, NJ, 1978

work page 1978

[5] [5]

Area minimizing hypersurfaces modulo p : a geometric free-boundary problem

Camillo De Lellis, Jonas Hirsch, Andrea Marchese, Luca Spolaor, and Salvatore Stuvard. Area minimizing hypersurfaces modulo p : a geometric free-boundary problem. J. Funct. Anal. , 290(12):Paper No. 111442, 111, 2026

work page 2026

[6] [6]

Elliptic regularization and partial regularity for motion by mean curvature

Tom Ilmanen. Elliptic regularization and partial regularity for motion by mean curvature. Mem. Amer. Math. Soc. , 108(520):x+90, 1994

work page 1994

[7] [7]

A general regularity theory for weak mean curvature flow

Kota Kasai and Yoshihiro Tonegawa. A general regularity theory for weak mean curvature flow. Calc. Var. Partial Differential Equations , 50(1-2):1--68, 2014

work page 2014

[8] [8]

The structure of stable codimension one integral varifolds near classical cones of density Q+1/2

Paul Minter. The structure of stable codimension one integral varifolds near classical cones of density Q+1/2 . Calc. Var. Partial Differential Equations , 63(1):Paper No. 5, 28, 2024

work page 2024