Singularities of mean curvature flow with bounded mean curvature and Morse index

Yongheng Han

REVIEW 3 major objections 5 minor 41 references

Singularities of mean curvature flow with bounded mean curvature and Morse index

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read For closed embedded mean curvature flows in dimensions 3 through 6, a first singularity forces either the mean curvature or the Morse index to blow up; in higher dimensions the singular set has Minkowski dimension at most n−7.

desk verdict A genuinely new multiplicity-one dichotomy for MCF with bounded mean curvature and bounded index, but the version on arXiv is conditional: the load-bearing two-sided pseudolocality estimate, Theorem 2.6, is stated without proof. read the letter →

arxiv 2501.05489 v3 pith:OO4G7CYP submitted 2025-01-09 math.DG

Yongheng Han This is my paper · ORCID

classification math.DG MSC 53E1053A1049Q20

keywords meancurvatureflowMorseindexmultiplicity-oneconjecturesingularitiesminimalconespseudolocalitystablehypersurfacesMinkowskidimension

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 proves that a closed smooth embedded mean curvature flow in $\mathbb{R}^{n+1}$ with uniformly bounded mean curvature and uniformly bounded Morse index cannot produce a first singular time when $3 \leq n \leq 6$: it must converge smoothly through that time. In all dimensions $n \geq 3$, the flow converges smoothly with multiplicity one to a limit hypersurface away from a singular set of Minkowski dimension at most $n-7$, and every blow-up limit is a stable minimal cone obtained with multiplicity one. The equivalent dichotomy is that at the first singular time, either the mean curvature or the Morse index (the number of negative eigenvalues of $\Delta+|A|^2$) must blow up. The interest is that bounded index and bounded mean curvature together rule out the stacked, multiple-sheet singular behavior that has been the main obstacle in the multiplicity-one problem.

What carries the argument

The load-bearing mechanism is the two-sided pseudolocality estimate, Theorem 2.6: if a time slice is nearly Euclidean in a ball and the whole flow has uniformly bounded mean curvature, then the second fundamental form is controlled in a small spacetime cylinder. Around this, the paper builds a weak compactness theory for hypersurfaces with bounded mean curvature, area ratio, and Morse index (Theorem 3.3), which gives $C^{1,\alpha}$ convergence away from a finite set of at most $I$ points where index concentrates, with $H^{n-2}(\operatorname{sing} M)=0$. The multiplicity-one argument then passes to the rescaled flow and studies the height difference between the top and bottom sheets of a putative multi-sheeted limit over a stable cone: the normalized height difference satisfies a parabolic equation close to the linearized shrinker operator $L = \Delta - \frac{1}{2}\langle x,\nabla\cdot\rangle + |A|^2 + \frac{1}{2}$, and parabolic Harnack estimates force a positive solution of $Lw=0$ on the cone. A positive solution of that sign would make the cone $L$-stable, which contradicts the strong Frankel theorem for shrinkers: no proper $F$-stationary rectifiable hypersurface with $H^{n-2}(\operatorname{sing} V)=0$ is $L$-stable. Hence the multiplicity must be one.

What would settle it

A concrete search: look for a smooth embedded mean curvature flow with uniformly bounded mean curvature whose time-zero slice has area ratio $1+o(1)$ in a ball, but whose second fundamental form exceeds $1/(\varepsilon r_0)$ inside the parabolic cylinder predicted by Theorem 2.6. Finding one would disprove the unproved pseudolocality estimate and collapse the proof's foundation; proving Theorem 2.6 would close the paper's only explicit gap.

Watch

Extended reading notes

Core claim

The paper establishes Theorem 0.4: for $n \geq 3$, let $M_t$ be a closed smooth embedded mean curvature flow in $\mathbb{R}^{n+1}$ on $[0,T)$ with $\sup |H| = \Lambda < \infty$ and $\sup_t \operatorname{index}(M_t) = I < \infty$. Then there is a limit hypersurface $\bar M_T$ and a subset $S \subset \bar M_T$ such that $M_t$ converges smoothly to $\bar M_T$ away from $S$ with multiplicity one, and $S$ has Minkowski dimension at most $n-7$. Consequently, for $3 \leq n \leq 6$, the flow does not blow up at time $T$. The proof also shows that for any point $p \in \bar M_T$ and any sequence $t_i \to T$, the rescaled surfaces $\frac{1}{\sqrt{T-t_i}}(M_{t_i}-p)$ converge, after subsequence, to a stable minimal cone with multiplicity one; in low dimensions the only such cone is a plane, so every point has Gaussian density one and is regular by the standard local regularity theorem for mean curvature flow.

Load-bearing premise

The proof depends on Theorem 2.6, a two-sided pseudolocality estimate for flows with uniformly bounded mean curvature, which is stated and used without proof; if that estimate is false, curvature could build up away from the presumed singular set and the convergence to a stable limit cone would break down.

Editorial extensions

If this is right

For $3 \leq n \leq 6$, any closed smooth embedded mean curvature flow that reaches a first singular time must have either unbounded mean curvature or unbounded Morse index; if both stay bounded, the flow remains smooth through that time.
For $n \geq 7$, under the same bounds the flow converges smoothly with multiplicity one to a limit hypersurface away from a singular set of Minkowski dimension at most $n-7$, and every blow-up limit is a stable minimal cone with multiplicity one.
In dimension 7 the singular set is discrete, and near each singular point the rescaled flow approaches a stable regular cone.
Quantitatively, the set where the parabolic regularity scale is below $r$ has volume at most $C(1+I)r^8$ in spacetime and $C(1+I)r^7$ at each time slice, giving finite $(n-7)$-dimensional Hausdorff measure for the time-slice singular set and finite $(n-5)$-dimensional measure for the spacetime singular set.
The theorem gives a clean dichotomy at the first singular time: either the mean curvature or the Morse index blows up, with no third possibility under the smooth embedded assumption.

Reading between the lines

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

The paper leaves Theorem 2.6, the two-sided pseudolocality estimate for bounded mean curvature, explicitly unproved; a reader should treat the theorem as conditional on that estimate, since all later compactness steps use it.
The codimension-7 bound matches the singular-set dimension for stable minimal hypersurfaces, suggesting that bounded index upgrades the singular behavior of a bounded-mean-curvature flow to the stable-minimal setting; sharpness could be probed with known non-flat stable minimal cones in dimensions $n \geq 7$.
The dichotomy raises a quantitative question the paper does not answer: if $|H|$ stays bounded at a singularity in dimensions 3 through 6, how fast must the Morse index grow, and is the growth rate determined by the singularity type?
The same height-difference mechanism might extend to flows with controlled index growth or to ambient manifolds with positive Ricci curvature, where the final Frankel-type obstruction would need a different form.

Share X Bluesky LinkedIn Reddit HN

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

The result is substantial and, if completed, would resolve a conditional form of Ilmanen's multiplicity-one problem in dimensions 3 through 6. The main theorem — bounded |H| and bounded Morse index imply smooth multiplicity-one convergence away from a set of Minkowski dimension at most n−7, and hence the dichotomy for 3 ≤ n ≤ 6 — is genuinely new. The weak compactness theorems (3.3 and 4.2) are useful tools in their own right, and the extension of Li–Wang's height-difference argument to the bounded-mean-curvature setting is a real step forward, not a routine transcription. I also see no circular reasoning: the proof derives the dichotomy from established compactness, regularity, and stability theorems, plus new technical inputs.

The soft spot is exactly where the stress-test note puts it. Theorem 2.6 is introduced with "We omit the proof here," and it is not a peripheral lemma. Lemma 6.5 uses Theorem 2.6 to convert a time-0 regularity-scale bound into a space-time regularity-scale bound on a full parabolic cylinder, and Theorem 6.8 then uses that to get the quantitative volume estimates and the Minkowski dimension bound in Theorem 0.4. Without a proof of Theorem 2.6, the singular-set dimension and Theorem 0.6 are conditional. The n ≤ 6 no-blow-up statement might survive through the density-one argument, but the paper as written does not verify the full theorem.

There are smaller gaps in the same direction. Lemma 5.7's compactness argument has an unstated case when the limit point lies on the singular set of the limit cone: varifold convergence does not by itself justify "smoothly converges near x∞," so the contradiction is not fully established. Lemma 5.3 asserts the uniform lower area ratio needed to form a refined sequence, and Lemma 5.10 asserts without proof that the relevant components remain in a fixed tubular neighborhood of the limit cone. These look fillable, but they are not just cosmetic. I would also flag that Theorem 6.8 is labeled with reference [3], but the statement is for Brakke flows and depends on Lemma 6.5 and Theorem 6.7; the attribution is at best imprecise and should be cleaned up.

My guess is that Theorem 2.6 is true and can be proved by adapting the proof of Theorem 2.5 with a scale-dependent cutoff and the mean curvature bound, but that is a guess. As submitted, the paper is a promising proof sketch with one central unproved input and a few technical assertions that need checking. It deserves peer review, not desk rejection, but the referee should expect a major revision. I would not cite the main theorem until Theorem 2.6 is proved.

Referee Report

3 major / 5 minor

Summary. The paper studies closed smooth embedded mean curvature flows in R^{n+1} for n ≥ 3 under two global bounds: a uniform L∞ bound on the mean curvature and a uniform bound on the Morse index of the time slices. The main theorem (Theorem 0.4) claims that, under these bounds, the flow converges smoothly with multiplicity one to a limit hypersurface away from a singular set of Minkowski dimension at most n−7; in particular, for 3 ≤ n ≤ 6 the flow does not blow up at the first singular time. The proof follows the Li-Wang strategy: a two-sided pseudolocality estimate, weak compactness theorems for hypersurfaces and flows, rescaling to stable minimal cones, and a multiplicity-one argument based on L-stability. Theorem 0.6 states quantitative estimates for the space-time singular set. The paper also states a corollary for n = 7 with a discrete singular set.

Significance. If the results are fully established, they would be a substantial advance: the n ≤ 6 no-blow-up statement under bounded mean curvature and bounded index directly addresses a conjecture related to Ilmanen's multiplicity-one conjecture, and the quantitative singular-set estimates for n ≥ 7 are in the spirit of Cheeger-Haslhofer-Naber. The paper is carefully organized around standard compactness, regularity, and stability tools, and it explicitly credits prior work through precise citations. However, the central proof currently depends on an unproved two-sided pseudolocality theorem (Theorem 2.6), with the manuscript stating 'We omit the proof here.' As a result, the main theorems should be regarded as conditional until that input is supplied or replaced.

major comments (3)

[Section 2, Theorem 2.6] Theorem 2.6 is a load-bearing input for the main theorems, but it is stated without proof: the text after the statement reads 'We omit the proof here.' This estimate is used in Lemma 6.5 to pass from the time-zero regularity scale r_{M0}(0) to the space-time regularity scale r_M(X) on a full parabolic cylinder, and Theorem 6.8 then uses Lemma 6.5 to obtain the volume estimates (6.11)-(6.12) and the Minkowski dimension bound in Theorem 0.4. The proved Theorem 2.5 does not cover Theorem 2.6: its proof uses the smallness of δ in an essential way (for example in (2.14)) and applies the Ecker-Huisken interior estimate (Lemma 2.1) only for positive times, so it gives no backward-in-time curvature control. Please provide a complete proof of Theorem 2.6, a precise reference, or a clear statement of which theorems are conditional on it.
[Section 5, Lemma 5.7] Lemma 5.7 defines the quantity s_C(x) only for x ∈ reg(C), but its conclusion is stated for all x ∈ (C ∩ B_R(0)) \ H(C,ε,R). The set H(C,ε,R) is a neighborhood of the low-curvature set S, not of the singular set of C, so points of sing(C) are included in the stated domain of the conclusion. The proof asserts that because x_j ∉ H(C_j,ε,R), 'C_j smoothly converges to C near x∞'; this is unjustified when x∞ ∈ sing(C), and the class C(N,n) contains singular cones. Lemma 5.8 (|T_N(C,ε,ζ,R)| = 0) and Lemma 5.9 use the conclusion of Lemma 5.7 for the limiting cones, so the thin-part argument is incomplete unless the singular set is handled separately or shown to be negligible.
[Section 5, Lemmas 5.3 and 5.9] The proof of Lemma 5.3 asserts, without argument, property (d) of the renormalized sequence, namely a uniform lower bound on the area ratio; this lower bound is part of the definition of a refined sequence and is used in Proposition 4.2. Lemma 5.9 also asserts convergence of the thin parts |T_N(Σ_{t_i},ε,ζ,R)| to |T_N(Σ∞,ε,ζ,R)| even though the convergence in Lemma 5.3 is only smooth away from sing(Σ∞) ∪ S_0, and points approaching the singular set or the time-dependent set e^{t/2}S_0 are not controlled. Please supply the missing estimates or state explicitly how the singular and exceptional sets are bypassed in this convergence.

minor comments (5)

[Section 5, Definition 5.4] The regularity scale is defined by 'sup_{y∈M∩B_r(y)} r|A|(y) ≤ 1'; the ball should be centered at the point x where the scale is being evaluated, not at the running point y.
[Section 5, equation (5.29)] The displayed inequality 'C_2(ε,S,T,x_0) < w_i(x,t) < C_1(ε,K,S_0,x_0) > 0' contains a typo; the intended statement is 0 < C_2 < w_i(x,t) < C_1.
[Section 2, Theorem 2.6] The statement of Theorem 2.6 begins 'For any r ∈ (0,1], T ≥ 1/2 and Λ > 0' but the hypotheses and conclusion use r_0; the notation should be aligned.
[Section 5, Lemma 5.7] The lemma says 'There exists ζ_0(R,N,ρ) > 0 with that for any C ∈ C(N,ρ)', even though no parameter ρ has been introduced in the statement; this should read C(N,n) and the dependence of ζ_0 should be stated consistently.
[Appendix A and Theorem 3.3] The proof of the index-zero case of Theorem 3.3 is delegated to 'repeat the proof of Theorem 2 in [32]' with no detailed adaptation to the bounded-mean-curvature setting; since this theorem is central to the paper, please either include the adapted proof or state precisely which arguments from [32] carry over unchanged.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation relies on established external compactness/regularity results and on a stated-but-unproved pseudolocality estimate that is not an input to itself.

full rationale

The paper's derivation chain is not circular. The main theorem follows from varifold compactness (Theorem 3.3), weak compactness of flows (Proposition 4.2), rescaling, and the multiplicity-one argument of Theorem 5.1, whose inputs are the bounded-mean-curvature and bounded-index hypotheses together with Huisken monotonicity, Schoen-Simon regularity, Sharp's index compactness, and Colding-Minicozzi stability results. These are quoted as external theorems, not as restatements of the target conclusion. No fitted parameters, data subsets, or quantities defined in terms of the claimed output appear. The only notable weakness is Theorem 2.6 in Section 2, introduced with: "Similarly, we also have the following two-sided pseudolocality theorem when H is uniformly bounded rather than small enough. We omit the proof here." This theorem is load-bearing for Lemma 6.5 and hence for the Minkowski-dimension estimate in Theorem 0.4. However, this is an unproved technical estimate, not a circular step: its hypotheses (area-ratio bound at time zero and uniform |H| ≤ Λ) do not already contain the desired regularity or singular-set dimension, and the argument is not replaced by a self-citation. The paper therefore contains a serious proof gap but no circularity.

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

The central claim rests on a chain of established results in geometric measure theory and on one new but unproved pseudolocality estimate (Theorem 2.6). No free parameters are fitted, and no new geometric or physical entities are postulated.

assumptions (6)

standard math Allard compactness and regularity theorems
Used in Section 1.1 to obtain varifold limits and local graph representations. Standard background in geometric measure theory.
standard math Schoen-Simon regularity for stable minimal hypersurfaces
Used in Section 3 (Theorem 3.1) and Appendix A to control the singular set of limits and to prove the stable case of the weak compactness theorem.
standard math Sharp's compactness and index lower semicontinuity
Used in Lemma 1.9 and Theorem 3.3 to transfer the bounded index of the sequence to the limit hypersurface.
standard math Colding-Minicozzi strong Frankel theorem
Used as Lemma 1.12 to assert every proper F-stationary rectifiable varifold in a large ball is L-unstable; this provides the contradiction in Theorem 5.1.
standard math Huisken's monotonicity formula and White's local regularity theorem
Used in Section 5.5 to compute the Gaussian density and to conclude that points with density one are regular space-time points.
ad hoc to paper Theorem 2.6, two-sided pseudolocality with bounded mean curvature
Stated in Section 2 with the proof omitted. It is a new estimate introduced for this paper that underlies the compactness of refined sequences and the multiplicity-one convergence. It is load-bearing and not proven.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Singularities of mean curvature flow with bounded mean curvature and Morse index." pith.science (2026). https://pith.science/paper/OO4G7CYP

@misc{pith2026250105489,
  author       = {Pith},
  title        = {Pith review of: Singularities of mean curvature flow with bounded mean curvature and Morse index},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OO4G7CYP}},
  note         = {Machine review of arXiv:2501.05489}
}

abstract

We study the multiplicity of the singularities of mean curvature flow with bounded mean curvature and Morse index. For $3\leq n\leq 6$, we show that either the mean curvature or the Morse index blows up at the first singular time for a closed smooth embedded mean curvature flow in $\mathbb{R}^{n+1}$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

41 extracted references · 37 canonical work pages

[3]

S. Aiex, S. McCurdy, and P. Minter, Quantitative Estimat es on the Singular Set of Minimal Hyper- surfaces with Bounded Index, J. Geom. Anal. 34(12) (2024) 372

work page 2024
[1]

Allard, On the ﬁrst variation of a varifold, Ann

W. Allard, On the ﬁrst variation of a varifold, Ann. of Math. 95(3) (1972) 417–491

work page 1972
[2]

Type II smoothing in mean curvature flow

S. Angenent, P. Daskalopoulos, and N. Sesum, Type II smoo thing in mean curvature ﬂow, arXiv:2108.08725 (2021)

work page Pith review arXiv 2021
[4]

Andrews, Noncollapsing in mean-convex mean curvatur e ﬂow, Geom

B. Andrews, Noncollapsing in mean-convex mean curvatur e ﬂow, Geom. Topol. 16(3) (2012) 1413– 1418

work page 2012
[5]

Bamler and B

R. Bamler and B. Kleiner, On the multiplicity one conject ure for mean curvature ﬂows of surfaces, arXiv:2312.02106(2023)

arXiv 2023
[6]

Brakke, The motion of a surface by its mean curvature, Mathematical Notes, Princeton University Press(1978)

K. Brakke, The motion of a surface by its mean curvature, Mathematical Notes, Princeton University Press(1978). 26 YONGHENG HAN

work page 1978
[7]

Chodosh, K.Choi, C

O. Chodosh, K.Choi, C. Mantoulidis, and F.Schulze, Revi siting generic mean curvature ﬂow in R3, arXiv:2409.01463(2024)

arXiv 2024
[8]

K. Choi, R. Haslhofer, and O. Hershkovits, Ancient low-e ntropy ﬂows, mean-convex neighborhoods, and uniqueness, Acta Math. 228(2) (2022) 217-301

work page 2022

Show all 41 references

[9]

Cheeger, R

J. Cheeger, R. Haslhofer, and A. Naber, Quantitative str atiﬁcation and the regularity of mean curvature ﬂow, Geom. Funct. Anal. 23(3) (2013) 828–847

2013
[10]

Colding and W

T. Colding and W. Minicozzi, Generic mean curvature ﬂow I; generic singularities, Ann. of Math. 175 (2012) 755–833

2012
[11]

Colding and W

T. Colding and W. Minicozzi, Uniqueness of blowups and /suppressLojasiewicz inequalities, Ann. of Math. 182 (2015) 221–285

2015
[12]

Colding and W

T. Colding and W. Minicozzi, The singular set of mean cur vature ﬂow with generic singularities, Invent. Math. 204(2) (2016) 443–471

2016
[13]

Colding and W

T. Colding and W. Minicozzi, A strong Frankel Theorem fo r shrinkers, arXiv:2306.08078(2023)

2023 arXiv
[14]

Cheeger and A

J. Cheeger and A. Naber, Quantitative stratiﬁcation an d the regularity of harmonic maps and min- imal currents, Comm. Pure Appl. Math. 66(6) (2013) 965–990

2013
[15]

Chodosh and F

O. Chodosh and F. Schulze, Uniqueness of asymptoticall y conical tangent ﬂows, Duke Math. J. 170(16) (2021) 3601–3657

2021
[16]

Chen and L

B. Chen and L. Yin, Uniqueness and pseudolocality theor ems of the mean curvature ﬂow, Commun. Anal. Geom. 15(3) (2007) 435–490

2007
[17]

Ecker and G

K. Ecker and G. Huisken, Interior estimates for hypersu rfaces moving by mean-curvarture, Invent. Math. 105(1) (1991) 547–569

1991
[18]

Guo and N

S. Guo and N. Sesum, Analysis of Vel´ azquez’s solution to the mean curvature ﬂow with a type II singularity, Commun. Partial Diﬀer. Equ. 43(2) (2018) 185–285

2018
[19]

Haslhofer and Bruce Kleiner, Mean curvature ﬂow of me an convex hypersurfaces, Comm

R. Haslhofer and Bruce Kleiner, Mean curvature ﬂow of me an convex hypersurfaces, Comm. Pure Appl. Math. 70(3) (2017) 511–546

2017
[20]

Huisken, Flow by mean curvature of convex surfaces into spheres, J

G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Diﬀer. Geom. 20(1) (1984) 237–266

1984
[21]

Huisken, Asymptotic behavior for singularities of t he mean curvature ﬂow, J

G. Huisken, Asymptotic behavior for singularities of t he mean curvature ﬂow, J. Diﬀer. Geom. 31(1) (1990) 285–299

1990
[22]

Huisken and C

G. Huisken and C. Sinestrari, Convexity estimates for m ean curvature ﬂow and singularities of mean convex surfaces, Acta Math. 183(1) (1999) 45–70

1999
[23]

Ilmanen, Elliptic regularization and partial regularity for motion by mean curvature,Mem

T. Ilmanen, Elliptic regularization and partial regularity for motion by mean curvature,Mem. Amer. Math. Soc. 108(520) (1994)

1994
[24]

Ilmanen, Lectures on mean curvature ﬂow and related e qua- tions,http://www.math.ethz.ch/ ilmanen/papers/pub.html,1995

T. Ilmanen, Lectures on mean curvature ﬂow and related e qua- tions,http://www.math.ethz.ch/ ilmanen/papers/pub.html,1995

1995
[25]

Ilmanen, A strong maximum principle for singular min imal hypersurfaces, Calc

T. Ilmanen, A strong maximum principle for singular min imal hypersurfaces, Calc. Var. Partial Diﬀer. Equ. 4(5) (1996) 443–467

1996
[26]

Li and B

H. Li and B. Wang, The extension problem of the mean curva ture ﬂow (I), Invent. Math. 218(3) (2019) 721–777

2019
[27]

Li and B

H. Li and B. Wang, On Ilmanen’s multiplicity-one conjec ture for mean curvature ﬂow with type-I mean curvature, J. Eur. Math. Soc. 24(1) (2022)

2022
[28]

Liu, Blow up of compact mean curvature ﬂow solutions w ith bounded mean curva- ture, arXiv:2403.16515(2024)

Z. Liu, Blow up of compact mean curvature ﬂow solutions w ith bounded mean curva- ture, arXiv:2403.16515(2024)

2024 arXiv
[29]

Naber and D

A. Naber and D. Valtorta, The singular structure and reg ularity of stationary varifolds, J. Eur. Math. Soc. 22(10) (2020) 3305–3382

2020
[30]

Sharp, Compactness of minimal hypersurfaces with bo unded index, J

B. Sharp, Compactness of minimal hypersurfaces with bo unded index, J. Diﬀer. Geom. 106(2) (2017) 317–339

2017
[31]

Simon, Lectures on geometric measure theory, Number 3 in Proceedings of the Centre for Math- ematical Analysis, Australian National University (1984)

L. Simon, Lectures on geometric measure theory, Number 3 in Proceedings of the Centre for Math- ematical Analysis, Australian National University (1984)

1984
[32]

Schoen and L

R. Schoen and L. Simon, Regularity of stable minimal hyp ersurfaces, Comm. Pure Appl. Math. 34(6) (1981) 741–797

1981
[33]

Stolarski, Existence of mean curvature ﬂow singular ities with bounded mean curvature, Duke Math

M. Stolarski, Existence of mean curvature ﬂow singular ities with bounded mean curvature, Duke Math. J. 172(7) (2023) 1235–1292

2023
[34]

Stolarski, On the structure of singularities of weak mean curvature ﬂows with mean curvature bounds, arXiv:2311.16262(2023)

M. Stolarski, On the structure of singularities of weak mean curvature ﬂows with mean curvature bounds, arXiv:2311.16262(2023). SINGULARITY OF MEAN CUR V ATURE FLOWS WITH BOUNDED MEAN CUR V A TURE AND MORSE INDEX 27

2023 arXiv
[35]

Sun and J

A. Sun and J. Xue, Generic mean curvature ﬂows with cylin drical singulari- ties, arXiv:2210.00419(2022)

2022 arXiv
[36]

Vel´azquez, Curvature blow-up in perturbations of minimal cones evolving by mean curvature ﬂow, Ann

J. Vel´azquez, Curvature blow-up in perturbations of minimal cones evolving by mean curvature ﬂow, Ann. Sc. norm. super. Pisa - Cl. sci. 21(4) (1994) 595–628

1994
[37]

Wang, Asymptotic structure of self-shrinkers, arXiv:1610.04904 (2016)

L. Wang, Asymptotic structure of self-shrinkers, arXiv:1610.04904 (2016)

2016 arXiv
[38]

White, Stratiﬁcation of minimal surfaces, mean curv ature ﬂows, and harmonic maps, J

B. White, Stratiﬁcation of minimal surfaces, mean curv ature ﬂows, and harmonic maps, J. Reine Angew. Math. 485(1997) 1–36

1997
[39]

White, The size of the singular set in mean curvature ﬂ ow of mean-convex sets, J

B. White, The size of the singular set in mean curvature ﬂ ow of mean-convex sets, J. Amer. Math. Soc. 13(3) (2000) 665–695

2000
[40]

White, The nature of singularities in mean curvature ﬂow of mean-convex sets, J

B. White, The nature of singularities in mean curvature ﬂow of mean-convex sets, J. Amer. Math. Soc. 16(1) (2003) 123–138

2003
[41]

White, A local regularity theorem for mean curvature ﬂow, Ann

B. White, A local regularity theorem for mean curvature ﬂow, Ann. of Math. 161(3) (2005) 1487– 1519. School of Mathmatical Science, University of Science and Te chnology of China, Hefei City, Anhui Province 230026 Email address : hyh2804@mail.ustc.edu.cn

2005

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [3]

S. Aiex, S. McCurdy, and P. Minter, Quantitative Estimat es on the Singular Set of Minimal Hyper- surfaces with Bounded Index, J. Geom. Anal. 34(12) (2024) 372

work page 2024

[2] [1]

Allard, On the ﬁrst variation of a varifold, Ann

W. Allard, On the ﬁrst variation of a varifold, Ann. of Math. 95(3) (1972) 417–491

work page 1972

[3] [2]

Type II smoothing in mean curvature flow

S. Angenent, P. Daskalopoulos, and N. Sesum, Type II smoo thing in mean curvature ﬂow, arXiv:2108.08725 (2021)

work page Pith review arXiv 2021

[4] [4]

Andrews, Noncollapsing in mean-convex mean curvatur e ﬂow, Geom

B. Andrews, Noncollapsing in mean-convex mean curvatur e ﬂow, Geom. Topol. 16(3) (2012) 1413– 1418

work page 2012

[5] [5]

Bamler and B

R. Bamler and B. Kleiner, On the multiplicity one conject ure for mean curvature ﬂows of surfaces, arXiv:2312.02106(2023)

arXiv 2023

[6] [6]

Brakke, The motion of a surface by its mean curvature, Mathematical Notes, Princeton University Press(1978)

K. Brakke, The motion of a surface by its mean curvature, Mathematical Notes, Princeton University Press(1978). 26 YONGHENG HAN

work page 1978

[7] [7]

Chodosh, K.Choi, C

O. Chodosh, K.Choi, C. Mantoulidis, and F.Schulze, Revi siting generic mean curvature ﬂow in R3, arXiv:2409.01463(2024)

arXiv 2024

[8] [8]

K. Choi, R. Haslhofer, and O. Hershkovits, Ancient low-e ntropy ﬂows, mean-convex neighborhoods, and uniqueness, Acta Math. 228(2) (2022) 217-301

work page 2022