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Passing through nondegenerate singularities in mean curvature flows

T0 review · 2 major / 3 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper proves that a nondegenerate cylindrical singularity of a mean curvature flow is isolated in spacetime, that the surrounding flow is mean convex, noncollapsing, and a smooth graph just before and just after the singular instant…

desk verdict A substantial new theorem about MCF through nondegenerate cylindrical singularities, but one unproved boundary-deformation hypothesis in Proposition 5.3(c) is load-bearing for the forward-time conclusions. read the letter →

arxiv 2501.16678 v1 pith:HAU2OP67 submitted 2025-01-28 math.DG math.APmath.GT

Ao Sun , Zhihan Wang , Jinxin Xue This is my paper · ORCID
classification math.DGmath.APmath.GT MSC 53E1035K9357R70
keywords meancurvatureflownondegeneratecylindricalsingularityBrakkeneckpinchdecayorderfrequencyfunctionMorsetheorynoncollapsing
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 aims to give a complete local answer to what happens when a mean curvature flow passes through a nondegenerate cylindrical singularity, the generic neck pinch in which a hypersurface collapses onto a generalized cylinder $C_{n,k}=S^{n-k}(\sqrt{2(n-k)})\times \mathbb{R}^k$. It claims that such a singularity is isolated in spacetime, that the flow near it is mean convex and noncollapsing, and that the moving surface is a smooth graph over the cylinder before the singular time, a cusped graph at the singular time, and a smooth graph over the dual cylinder afterwards. It further claims that the topology change is exactly the change of level sets near a Morse critical point, namely an $(n-k)$-surgery. Because the result holds for every dimension $n\ge 2$ and every cylinder type $C_{n,k}$, it supplies a canonical, parameter-free picture of the surgery that a mean curvature flow performs at a neck pinch.

What carries the argument

The engine is a new weighted $L^2$-distance monotonicity on the rescaled flow. Define $d_{n,k}(\Sigma)^2=\int \mathrm{dist}_{n,k}(X)^2 e^{-|X|^2/4}\,d\Sigma$, where $\mathrm{dist}_{n,k}$ is a regularized signed distance to $C_{n,k}$; a non-concentration estimate shows that the far-away part of this integral is controlled by its initial value. From it the paper defines the decay order $N_{n,k}(\tau;M)=\log\big(d_{n,k}(M(\tau))/d_{n,k}(M(\tau+1))\big)$, a discrete analogue of Almgren's frequency function, and proves a discrete almost-monotonicity: the decay order either drops by a definite amount or stabilizes near an eigenvalue of the Jacobi operator $-L_{n,k}$ on the cylinder. Since the small eigenvalues of $-L_{n,k}$ correspond exactly to the unstable modes (translation in the sphere and in the $\mathbb{R}^k$ directions, plus the quadratic Hermite modes), the stabilization identifies which linear mode dominates the flow. That identification, combined with a classification of noncollapsing ancient asymptotically cylindrical flows as either shrinking cylinders or bowl solitons times $\mathbb{R}^{k-1}$, yields the dichotomy of blow-up models from which the isolatedness conclusion and the post-singular graphical description follow.

What would settle it

Take a concrete nondegenerate neck pinch, such as the rotationally symmetric examples of [AV97], track the boundary curves $\mathrm{spt}\,M(t)\cap\partial B_{r_0}$ through the singular instant, and check whether a smooth monotonic family $\Gamma_t$ joining them to the dual-cylinder slices $(S^{n-k}(r_0')\times\mathbb{R}^k)\cap\partial B_{r_0}$ exists; if the boundary data folds back on itself or cannot be joined monotonically, the hypothesis of Proposition 5.3(c) fails and the noncollapsing argument — and with it the forward-time classification — does not go through.

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Extended reading notes

Core claim

The central claim is Theorem 1.1. For a unit-regular cyclic mod 2 Brakke flow $t\mapsto M(t)$ in $\mathbb{R}^{n+1}$ with a nondegenerate cylindrical singularity modeled by $C_{n,k}$ at the spacetime origin, the paper asserts that $(0,0)$ is the only singularity in a parabolic neighborhood $Q_{r_0}\times[-t_0,t_0]$; that the flow there is mean convex and noncollapsing; that for $t<0$ the surface is a $C^\infty$ graph over $C_{n,k}$; that at $t=0$ it is a graph with the universal cusp profile $u(\theta,y)=\sqrt{2(n-k)}\,\frac{|y|}{2\sqrt{-\log|y|}}\,(1+o_y(1))-\sqrt{2(n-k)}$; that for $t>0$ it is a smooth graph over the dual cylinder $C^*_{n,k}(r_0)=\mathbb{R}^{n-k+1}\times S^{k-1}(r_0)$; and that the topology change is an $(n-k)$-surgery, identical to the level-set transition near a Morse critical point of index $n-k+1$.

Load-bearing premise

The proof that the flow is noncollapsing after the singular time, and through that the classification of all possible blow-up shapes, assumes that the boundary traces of the flow on a fixed sphere can be smoothly and monotonically deformed into the boundary traces of the dual cylinder; the paper calls this “easy to check” while conjecturing it can be dropped, and if no such deformation exists the noncollapsing step is unsupported.

Editorial extensions

If this is right

  • Corollary 1.3: a mean curvature flow whose only singularities are nondegenerate cylindrical and spherical ones is unique, has only finitely many singularities, and its spacetime track admits a Morse function whose index-$(n-k+1)$ critical points are in one-to-one correspondence with the singularities modeled by $C_{n,k}$.
  • Corollary 1.5: if the flow starts from a closed $k$-convex hypersurface and satisfies the nondegeneracy condition, the enclosed domain carries a Morse function with no critical points of index $0,1,\dots,n-k+1$, so the domain is obtained from standard balls by attaching only handles of indices $1$ through $k-1$.
  • Corollaries 1.7 and 1.8: the Betti numbers of the initial hypersurface force a lower bound on the number of nondegenerate singularities of each cylinder type, because each topology change is now a completely understood surgery.
  • The singular-time profile is universal: at the singular instant the surface is a graph over $C_{n,k}$ whose deviation from the cylinder is $\sqrt{2(n-k)}\,|y|/(2\sqrt{-\log|y|})$ near the spine, with no dependence on the particular flow.
  • The surgery performed by the flow is canonical: the graphical descriptions before and after the singular time leave no freedom to choose when or where to cut, unlike earlier surgery constructions.

Reading between the lines

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

  • If the paper's Conjecture 1.4 is settled, the theorem becomes a 'missing handle' principle: the spacetime track of a generic mean convex flow would have a handle decomposition with no $n$-handles or $(n+1)$-handles, so standard balls would be the only building blocks.
  • The decay-order machinery is not restricted to nondegenerate singularities, and the paper announces a companion study of degenerate ones; a concrete test is to compute $N_{n,k}(\tau)$ in the rotationally symmetric neck-pinch examples of [AV97], where the cusp profile is explicit, and check that the limiting decay order lands on the predicted eigenvalue.
  • The cusp profile yields a quantitative prediction that numerical simulation could check: at the singular instant the neck radius should close like $|y|/(2\sqrt{-\log|y|})$ as a function of distance from the spine, independently of the initial shape.
  • Isolatedness of each nondegenerate singularity is exactly the local input needed to run surgeries sequentially, so a suitable a priori bound on the number of singular events could turn the local description into a global decomposition of any flow satisfying the nondegeneracy condition.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 3 minor

Summary. The paper studies unit-regular cyclic mod 2 Brakke flows with a nondegenerate cylindrical singularity modeled on C_{n,k} at a point. The main result, Theorem 1.1, claims that such a singularity is isolated in a parabolic neighborhood, that the flow is mean convex and noncollapsing there, that the flow is graphical over the cylinder before and at the singular time (with a cusp profile |y|/(2 sqrt(-log|y|)) at the singular time), that after the singular time it is a graph over the dual cylinder, and that the topology change is an (n-k)-surgery. The proof introduces an L^2-distance monotonicity formula and a discrete decay order, and combines them with pseudolocality, elliptic regularization, White regularity, and the Du-Zhu classification of noncollapsing ancient flows. The backward-time statements are taken from prior work of the first and third authors, and the forward-time statements are derived from a new classification of blow-up models.

Significance. If the main theorem holds, the paper provides a complete local description of the geometry and topology change near nondegenerate cylindrical singularities for arbitrary n and k, including the mean-convex-neighborhood property and surgery description. The new L^2-distance monotonicity (Corollary 3.3) and the discrete almost-monotonicity of the decay order (Corollary 3.7) are original tools with potential applications to other singularities. The paper also derives several global corollaries (Corollaries 1.3-1.8) about uniqueness, finiteness, and handle decompositions under the generic-singularity assumption. The analytic arguments are detailed and use standard machinery; the appendices provide useful technical lemmas. No machine-checked proofs are included, but the derivations are presented in a verifiable manner.

major comments (2)
  1. [Section 5.2, Proposition 5.3(c)] Proposition 5.3(c) postulates a smooth monotonic boundary deformation {Gamma_t}_{t>=0} with Gamma_t = spt M_t cap partial B_{r0} for t in [0,T] and Gamma_t = (S^{n-k}(r0') x R^k) cap partial B_{r0} for t >= T+1, and the paper asserts that this deformation exists and is 'easy to check', while also conjecturing that it can be dropped. No construction of Gamma_t is given for the specific nondegenerate-singularity flow of Theorem 1.1. This hypothesis is load-bearing: it defines the boundary portion of the minimizers N_lambda in the elliptic-regularization proof of Proposition 5.3(iii), and the smoothness/monotonicity of Gamma_t at the top and side boundaries is what yields the uniform two-sided bounds on Z*/H and Z_*/H that pass to the limit as lambda -> infinity. Without the noncollapsing conclusion (iii), Theorem 4.2 cannot invoke the Du-Zhu classification (Theorem 2.10), so the forward-time isolatedness (i) and the graphical/morphism description (vii) of Theorem 1.1 are unsupported.
  2. [Section 5.2, Proposition 5.3(c); Theorem 1.1(iv)] The claimed deformation Gamma_t is not immediate from the results proved earlier. Theorem 1.1(iv) controls spt M(t) cap partial Q_{r0} (the boundary of the product cylinder Q_{r0} = B^{n-k+1}_{r0} x B^k_{r0}), whereas Proposition 5.3(c) requires boundary data on the Euclidean sphere partial B_{r0} cap Omega. These are different boundaries, and no argument is given for how the flow's known boundary behavior on partial Q_{r0} yields a smooth monotone family Gamma_t on partial B_{r0} connecting the pre-singular boundary data to the dual-cylinder cross-section. Until such a construction is supplied, the forward-time noncollapsing conclusion (Theorem 1.1(iii)) and the classification of blow-up models in Theorem 4.2 remain conditional on an unproved assumption.
minor comments (3)
  1. [Throughout] The text contains numerous OCR-type artifacts in the displayed text and references (e.g., 'P ASSING', 'V A TURE', 'W ANG', 'f ˜A¼r'), which should be cleaned before publication.
  2. [Theorem 1.1(iv)] The phrase 'tubular neighborhood U(t) of {0} x S^{k-1}_{r0} in partial Q_{r0} cap C^*_{n,k}(r0)' is ambiguous because partial Q_{r0} cap C^*_{n,k}(r0) is not a manifold with boundary; please clarify the intended topology.
  3. [Section 5.2, Proposition 5.3] The paper's own remark after Proposition 5.3 acknowledging that assumption (c) is conjectural should be highlighted in the introduction as a limitation, since it currently appears only in the middle of the proof.

Circularity Check

0 steps flagged · score 2.0 of 10

A self-contained derivation with no fitted inputs; the main caveat is an unproved auxiliary deformation in Proposition 5.3, which is a gap rather than a circularity.

full rationale

The claimed derivation is not circular. The backward-time items of Theorem 1.1 are imported from [SX22], prior work by two of the authors, but that prior normal-form theorem is an independent result whose stated assumptions (a rescaled mean curvature flow converging to a cylinder with finite entropy) do not include Theorem 1.1; nondegeneracy is an input, not a conclusion. The new forward machinery (L2-distance monotonicity, decay order, discrete almost-monotonicity) is derived from the Brakke-flow equations without fitting parameters and without renaming known results as new predictions. Theorem 4.2 invokes the external Du-Zhu classification only after mean convexity and noncollapsing are established via elliptic regularization, not as a substitute for those facts. The sentence in the paper that Proposition 5.3 implies items (ii) and (iii) 'based on Theorem 1.1 (iv)-(vi)' is not circular because (iv)-(vi) are proved in Section 5.1 from pseudolocality without using (ii) or (iii). The only flagged limitation is Proposition 5.3(c), where a smooth monotonic boundary deformation Gamma_t is asserted without construction and the paper itself says 'The technical assumption (c) is only used to prove (iii), and we conjecture that it can be dropped.' This is load-bearing for forward-time noncollapsing and hence for the application of Theorem 4.2, but it is an omitted hypothesis or possible gap, not an identity between input and output, and not a fitted parameter relabeled as a prediction. No specific equation or conclusion reduces by construction to its own assumption, so no circular step is exhibited; the score of 2 reflects the self-citation reliance and the unproved auxiliary deformation rather than any circularity.

Assumptions & free parameters 0 free parameters · 7 assumptions · 1 invented entities

No free parameters are fitted to data. The proof depends on several prior theorems, listed as axioms, and on one technical deformation assumption introduced for this paper. The new monotonicity formula is derived from Brakke flow inequalities and the first variation formula.

assumptions (7)
  • standard math Huisken's monotonicity formula and Gaussian density theory for Brakke flows
    Invoked in Section 2.2 to define blow-ups, density, and rescaled flows, and used throughout Section 3.
  • standard math White's epsilon regularity and stratification theorems for Brakke flows
    Used in Section 4 to conclude isolatedness from the blow-up classification and in Section 5.2 for regularity.
  • domain assumption Normal form theorem for nondegenerate cylindrical singularities from [SX22, Theorem 2.5]
    Defines nondegeneracy in Definition 2.6 and supplies the backward-time graphical asymptotics used throughout; proved by the same authors in prior work.
  • domain assumption Du-Zhu classification of ancient noncollapsed asymptotically cylindrical flows [DZ22, Theorems 2.9-2.10]
    Used in Theorem 4.2 to force limiting blow-up models to be cylinders or bowl solitons times R^{k-1}; an external major theorem in the same research program.
  • domain assumption The given weak flow is a unit-regular cyclic mod 2 Brakke flow
    Theorem 1.1 is stated for this class; it is the class produced by elliptic regularization and used for uniqueness and regularity arguments.
  • ad hoc to paper Existence of smooth monotonic boundary deformation Gamma_t in Proposition 5.3(c)
    Used only for noncollapsing; asserted to be checkable but not proved, and the authors conjecture it can be dropped. This is the weakest axiom in the proof.
  • domain assumption Mean-convex and noncollapsing properties of the flow
    Theorem 4.2 assumes these properties, and Proposition 5.3 proves them for the actual flow; legitimate once the proposition is accepted.
invented entities (1)
  • decay order N_{n,k}(tau;M)
    purpose: A discrete parabolic analogue of Almgren's frequency function; it measures the exponential rate at which a rescaled flow converges to the cylinder and is used to control blow-up models.
    Defined in equation (3.18). It is a formal analytic quantity with no falsifiable handle outside the paper, not a physical postulate.

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Pith. "Pith review of Passing through nondegenerate singularities in mean curvature flows." pith.science (2026). https://pith.science/paper/HAU2OP67

@misc{pith2026250116678,
  author       = {Pith},
  title        = {Pith review of: Passing through nondegenerate singularities in mean curvature flows},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HAU2OP67}},
  note         = {Machine review of arXiv:2501.16678}
}
abstract

In this paper, we study the properties of nondegenerate cylindrical singularities of mean curvature flow. We prove they are isolated in spacetime and provide a complete description of the geometry and topology change of the flow passing through the singularities. Particularly, the topology change agrees with the level sets change near a critical point of a Morse function, which is the same as performing surgery. The proof is based on a new $L^2$-distance monotonicity formula, which allows us to derive a discrete almost monotonicity of the ``decay order", a discrete mean curvature flow analog to Almgren's frequency function.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Regularity of cylindrical singular sets of mean curvature flow

    math.DG 2025-09 conditional novelty 7.0 of 10

    Degenerate cylindrical singular sets of mean curvature flow are locally contained in C^{2,α} submanifolds, with curvature determined by the flow's asymptotic profile.

Reference graph

Works this paper leans on

92 extracted references · 52 canonical work pages · cited by 1 Pith paper

  1. [1]

    Angenent, and Yoshikazu Giga

    Steven Altschuler, Sigurd B. Angenent, and Yoshikazu Giga. Mean curvature flow through singularities for surfaces of rotation. J. Geom. Anal. , 5(3):293--358, 1995

    1995

  2. [2]

    Guadalupe Reyes, and Walcy Santos

    Hil\'ario Alencar, Abd\^enago Barros, Oscar Palmas, J. Guadalupe Reyes, and Walcy Santos. O (m) O (n) -invariant minimal hypersurfaces in R^ m+n . Ann. Global Anal. Geom. , 27(2):179--199, 2005

    2005

  3. [3]

    Unique asymptotics of ancient convex mean curvature flow solutions

    Sigurd Angenent, Panagiota Daskalopoulos, and Natasa Sesum. Unique asymptotics of ancient convex mean curvature flow solutions. J. Differential Geom. , 111(3):381--455, 2019

    2019

  4. [4]

    S. B. Angenent, T. Ilmanen, and J. J. L. Vel\' a zquez. Fattening from smooth initial data in mean curvature flow. Preprint , 2017

    2017

  5. [5]

    Optimal isoperimetric inequalities

    Fred Almgren. Optimal isoperimetric inequalities. Indiana University mathematics journal , 35(3):451--547, 1986

    1986

  6. [6]

    Non-collapsing in fully non-linear curvature flows

    Ben Andrews, Mat Langford, and James McCoy. Non-collapsing in fully non-linear curvature flows. In Annales de l'Institut Henri Poincar \'e C, Analyse non lin \'e aire , volume 30, pages 23--32. Elsevier, 2013

    2013

  7. [7]

    Noncollapsing in mean-convex mean curvature flow

    Ben Andrews. Noncollapsing in mean-convex mean curvature flow. Geom. Topol. , 16(3):1413--1418, 2012

    2012

  8. [8]

    Formal asymptotic expansions for symmetric ancient ovals in mean curvature flow

    Sigurd Angenent. Formal asymptotic expansions for symmetric ancient ovals in mean curvature flow. Netw. Heterog. Media , 8(1):1--8, 2013

    2013

Show all 92 references
  1. [9]

    S. B. Angenent and J. J. L. Vel\' a zquez. Degenerate neckpinches in mean curvature flow. J. Reine Angew. Math. , 482:15--66, 1997

    1997

  2. [10]

    Translating surfaces of the non-parametric mean curvature flow with prescribed contact angle

    Steven J Altschuler and Lang F Wu. Translating surfaces of the non-parametric mean curvature flow with prescribed contact angle. Calculus of Variations and Partial Differential Equations , 2(1):101--111, 1994

    1994

  3. [11]

    Uniqueness of convex ancient solutions to mean curvature flow in R^3

    Simon Brendle and Kyeongsu Choi. Uniqueness of convex ancient solutions to mean curvature flow in R^3 . Invent. Math. , 217(1):35--76, 2019

    2019

  4. [12]

    Uniqueness of convex ancient solutions to mean curvature flow in higher dimensions

    Simon Brendle and Kyeongsu Choi. Uniqueness of convex ancient solutions to mean curvature flow in higher dimensions. Geom. Topol. , 25(5):2195--2234, 2021

    2021

  5. [13]

    Mean curvature flow with surgery of mean convex surfaces in R^3

    Simon Brendle and Gerhard Huisken. Mean curvature flow with surgery of mean convex surfaces in R^3 . Invent. Math. , 203(2):615--654, 2016

    2016

  6. [14]

    Parabolic frequency for the mean curvature flow

    Julius Baldauf, Pak Tung Ho, and Tang-Kai Lee. Parabolic frequency for the mean curvature flow. Int. Math. Res. Not. IMRN , (10):8122--8136, 2024

    2024

  7. [15]

    On the multiplicity one conjecture for mean curvature flows of surfaces

    Richard H Bamler and Bruce Kleiner. On the multiplicity one conjecture for mean curvature flows of surfaces. arXiv preprint arXiv:2312.02106 , 2023

    2023 arXiv

  8. [16]

    Bamler and Bruce Kleiner

    Richard H. Bamler and Bruce Kleiner. Ricci flow and diffeomorphism groups of 3-manifolds. J. Amer. Math. Soc. , 36(2):563--589, 2023

    2023

  9. [17]

    Bamler, Chao Li, and Christos Mantoulidis

    Richard H. Bamler, Chao Li, and Christos Mantoulidis. Decomposing 4-manifolds with positive scalar curvature. Adv. Math. , 430:Paper No. 109231, 17, 2023

    2023

  10. [18]

    A local noncollapsing estimate for mean curvature flow

    Simon Brendle and Keaton Naff. A local noncollapsing estimate for mean curvature flow. arXiv preprint arXiv:2103.15641 , 2021

    2021 arXiv

  11. [19]

    Kenneth A. Brakke. The Motion of a Surface by Its Mean Curvature. (MN-20) . Princeton University Press, Princeton, 1978

    1978

  12. [20]

    A sharp bound for the inscribed radius under mean curvature flow

    Simon Brendle. A sharp bound for the inscribed radius under mean curvature flow. Invent. Math. , 202(1):217--237, 2015

    2015

  13. [21]

    Embedded self-similar shrinkers of genus 0

    Simon Brendle. Embedded self-similar shrinkers of genus 0. Ann. of Math. (2) , 183(2):715--728, 2016

    2016

  14. [22]

    Manifolds with 1/4 -pinched curvature are space forms

    Simon Brendle and Richard Schoen. Manifolds with 1/4 -pinched curvature are space forms. J. Amer. Math. Soc. , 22(1):287--307, 2009

    2009

  15. [23]

    A topological property of asymptotically conical self-shrinkers of small entropy

    Jacob Bernstein and Lu Wang. A topological property of asymptotically conical self-shrinkers of small entropy. Duke Math. J. , 166(3):403--435, 2017

    2017

  16. [24]

    Closed hypersurfaces of low entropy in R^4 are isotopically trivial

    Jacob Bernstein and Lu Wang. Closed hypersurfaces of low entropy in R^4 are isotopically trivial. Duke Math. J. , 171(7):1531--1558, 2022

    2022

  17. [25]

    Mean curvature flow with generic initial data

    Otis Chodosh, Kyeongsu Choi, Christos Mantoulidis, and Felix Schulze. Mean curvature flow with generic initial data. Invent. Math. , 237(1):121--220, 2024

    2024

  18. [26]

    Mean curvature flow with generic low-entropy initial data

    Otis Chodosh, Kyeongsu Choi, Christos Mantoulidis, and Felix Schulze. Mean curvature flow with generic low-entropy initial data. Duke Math. J. , 173(7):1269--1290, 2024

    2024

  19. [27]

    Mean curvature flow with generic initial data ii

    Otis Chodosh, Kyeongsu Choi, and Felix Schulze. Mean curvature flow with generic initial data ii. arXiv preprint arXiv:2302.08409 , 2023

    2023 arXiv

  20. [28]

    Mean curvature flow with generic low-entropy initial data ii (2023)

    Otis Chodosh, Kyeongsu Choi, and Felix Schulze. Mean curvature flow with generic low-entropy initial data ii (2023). Preprint available at https://arxiv. org/abs/2309.03856 , 2023

    2023 arXiv

  21. [29]

    Classification of bubble-sheet ovals in r4

    B Choi, P Daskalopoulos, W Du, R Haslhofer, and N Sesum. Classification of bubble-sheet ovals in r4. arXiv preprint arXiv:2209.04931 , 2022

    2022 arXiv

  22. [30]

    Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations

    Yun Gang Chen, Yoshikazu Giga, and Shun'ichi Goto. Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations. Journal of differential geometry , 33(3):749--786, 1991

    1991

  23. [31]

    A note on the selfsimilarity of limit flows

    Beomjun Choi, Robert Haslhofer, and Or Hershkovits. A note on the selfsimilarity of limit flows. Proc. Amer. Math. Soc. , 149(3):1239--1245, 2021

    2021

  24. [32]

    Ancient low-entropy flows, mean-convex neighborhoods, and uniqueness

    Kyeongsu Choi, Robert Haslhofer, and Or Hershkovits. Ancient low-entropy flows, mean-convex neighborhoods, and uniqueness. Acta Math. , 228(2):217--301, 2022

    2022

  25. [33]

    Classification of noncollapsed translators in R^4

    Kyeongsu Choi, Robert Haslhofer, and Or Hershkovits. Classification of noncollapsed translators in R^4 . Camb. J. Math. , 11(3):563--698, 2023

    2023

  26. [34]

    Ancient asymptotically cylindrical flows and applications

    Kyeongsu Choi, Robert Haslhofer, Or Hershkovits, and Brian White. Ancient asymptotically cylindrical flows and applications. Invent. Math. , 229(1):139--241, 2022

    2022

  27. [35]

    Rigidity of generic singularities of mean curvature flow

    Tobias Holck Colding, Tom Ilmanen, and William P Minicozzi. Rigidity of generic singularities of mean curvature flow. Publications math \'e matiques de l'IH \'E S , 121(1):363--382, 2015

    2015

  28. [36]

    The round sphere minimizes entropy among closed self-shrinkers

    Tobias Holck Colding, Tom Ilmanen, William P Minicozzi II, and Brian White. The round sphere minimizes entropy among closed self-shrinkers. Journal of Differential Geometry , 95(1):53--69, 2013

    2013

  29. [37]

    Generic mean curvature flow i; generic singularities

    Tobias H Colding and William P Minicozzi. Generic mean curvature flow i; generic singularities. Annals of mathematics , pages 755--833, 2012

    2012

  30. [38]

    Minicozzi, II

    Tobias Holck Colding and William P. Minicozzi, II. Uniqueness of blowups and ojasiewicz inequalities. Ann. of Math. (2) , 182(1):221--285, 2015

    2015

  31. [39]

    The singular set of mean curvature flow with generic singularities

    Tobias Holck Colding and William P Minicozzi. The singular set of mean curvature flow with generic singularities. Inventiones mathematicae , 204(2):443--471, 2016

    2016

  32. [40]

    Minicozzi, II

    Tobias Holck Colding and William P. Minicozzi, II. Regularity of the level set flow. Comm. Pure Appl. Math. , 71(4):814--824, 2018

    2018

  33. [41]

    Minicozzi, II

    Tobias Holck Colding and William P. Minicozzi, II. Parabolic frequency on manifolds. Int. Math. Res. Not. IMRN , (15):11878--11890, 2022

    2022

  34. [42]

    Genus one singularities in mean curvature flow

    Adrian Chun-Pong Chu and Ao Sun. Genus one singularities in mean curvature flow. arXiv preprint arXiv:2308.05923 , 2023

    2023

  35. [43]

    Uniqueness and pseudolocality theorems of the mean curvature flow

    Bing-Long Chen and Le Yin. Uniqueness and pseudolocality theorems of the mean curvature flow. Comm. Anal. Geom. , 15(3):435--490, 2007

    2007

  36. [44]

    On uniqueness and nonuniqueness of ancient ovals

    Wenkui Du and Robert Haslhofer. On uniqueness and nonuniqueness of ancient ovals. arXiv preprint arXiv:2105.13830 , 2021

    2021 arXiv

  37. [45]

    J. M. Daniels-Holgate. Approximation of mean curvature flow with generic singularities by smooth flows with surgery. Adv. Math. , 410:Paper No. 108715, 42, 2022

    2022

  38. [46]

    Hearing the shape of ancient noncollapsed flows in R^4

    Wenkui Du and Robert Haslhofer. Hearing the shape of ancient noncollapsed flows in R^4 . Communications on Pure and Applied Mathematics , 77(1):543--582, 2024

    2024

  39. [47]

    Spectral quantization for ancient asymptotically cylindrical flows

    Wenkui Du and Jingze Zhu. Spectral quantization for ancient asymptotically cylindrical flows. arXiv preprint arXiv:2211.02595 , 2022

    2022 arXiv

  40. [48]

    Interior estimates for hypersurfaces moving by mean curvature

    Klaus Ecker and Gerhard Huisken. Interior estimates for hypersurfaces moving by mean curvature. Inventiones mathematicae , 105(1):547--569, 1991

    1991

  41. [49]

    Motion of level sets by mean curvature

    Lawrence C Evans and Joel Spruck. Motion of level sets by mean curvature. i. Journal of Differential Geometry , 33(3):635--681, 1991

    1991

  42. [50]

    On the mean convexity of a space-and-time neighborhood of generic singularities formed by mean curvature flow

    Zhou Gang. On the mean convexity of a space-and-time neighborhood of generic singularities formed by mean curvature flow. J. Geom. Anal. , 31(10):9819--9890, 2021

    2021

  43. [51]

    On the dynamics of formation of generic singularities of mean curvature flow

    Zhou Gang. On the dynamics of formation of generic singularities of mean curvature flow. J. Funct. Anal. , 282(12):Paper No. 109458, 73, 2022

    2022

  44. [52]

    Blaine Lawson, Jr

    Mikhael Gromov and H. Blaine Lawson, Jr. The classification of simply connected manifolds of positive scalar curvature. Ann. of Math. (2) , 111(3):423--434, 1980

    1980

  45. [53]

    Mean curvature in the light of scalar curvature

    Misha Gromov. Mean curvature in the light of scalar curvature. Ann. Inst. Fourier (Grenoble) , 69(7):3169--3194, 2019

    2019

  46. [54]

    Precise asymptotics near a generic s1 r3 singularity of mean curvature flow

    Zhou Gang and Shengwen Wang. Precise asymptotics near a generic s1 r3 singularity of mean curvature flow. Nonlinear Analysis , 251:113679, 2025

    2025

  47. [55]

    Mean curvature flow of mean convex hypersurfaces

    Robert Haslhofer and Bruce Kleiner. Mean curvature flow of mean convex hypersurfaces. Communications on Pure and Applied Mathematics , 70(3):511--546, 2017

    2017

  48. [56]

    Mean curvature flow with surgery

    Robert Haslhofer and Bruce Kleiner. Mean curvature flow with surgery. Duke Math. J. , 166(9):1591--1626, 2017

    2017

  49. [57]

    Area minimizing hypersurfaces with isolated singularities

    Robert Hardt and Leon Simon. Area minimizing hypersurfaces with isolated singularities. Journal f\"ur die reine und angewandte Mathematik , 362:102--129, 1985

    1985

  50. [58]

    Convexity estimates for mean curvature flow and singularities of mean convex surfaces

    Gerhard Huisken and Carlo Sinestrari. Convexity estimates for mean curvature flow and singularities of mean convex surfaces. Acta mathematica , 183(1):45--70, 1999

    1999

  51. [59]

    Mean curvature flow with surgeries of two-convex hypersurfaces

    Gerhard Huisken and Carlo Sinestrari. Mean curvature flow with surgeries of two-convex hypersurfaces. Invent. Math. , 175(1):137--221, 2009

    2009

  52. [60]

    Asymptotic behavior for singularities of the mean curvature flow

    Gerhard Huisken. Asymptotic behavior for singularities of the mean curvature flow. Journal of Differential Geometry , 31(1):285--299, 1990

    1990

  53. [61]

    Local and global behaviour of hypersurfaces moving by mean curvature

    Gerhard Huisken. Local and global behaviour of hypersurfaces moving by mean curvature. In Differential geometry: partial differential equations on manifolds ( L os A ngeles, CA , 1990) , volume 54, Part 1 of Proc. Sympos. Pure Math. , pages 175--191. Amer. Math. Soc., Providen...

    1990

  54. [62]

    Nonfattening of mean curvature flow at singularities of mean convex type

    Or Hershkovits and Brian White. Nonfattening of mean curvature flow at singularities of mean convex type. Communications on Pure and Applied Mathematics , 73(3):558--580, 2020

    2020

  55. [63]

    Avoidance for set-theoretic solutions of mean-curvature-type flows

    Or Hershkovits and Brian White. Avoidance for set-theoretic solutions of mean-curvature-type flows. Comm. Anal. Geom. , 31(1):31--67, 2023

    2023

  56. [64]

    Generalized flow of sets by mean curvature on a manifold

    Tom Ilmanen. Generalized flow of sets by mean curvature on a manifold. Indiana University mathematics journal , pages 671--705, 1992

    1992

  57. [65]

    Elliptic regularization and partial regularity for motion by mean curvature , volume 520

    Tom Ilmanen. Elliptic regularization and partial regularity for motion by mean curvature , volume 520. American Mathematical Soc., 1994

    1994

  58. [66]

    On short time existence for the planar network flow

    Tom Ilmanen, Andr\' e Neves, and Felix Schulze. On short time existence for the planar network flow. J. Differential Geom. , 111(1):39--89, 2019

    2019

  59. [67]

    Fattening in mean curvature flow

    Tom Ilmanen and Brian White. Fattening in mean curvature flow. arXiv preprint arXiv:2406.18703 , 2024

    2024 arXiv

  60. [68]

    Self-shrinkers whose asymptotic cones fatten

    Daniel Ketover. Self-shrinkers whose asymptotic cones fatten. arXiv preprint arXiv:2407.01240 , 2024

    2024 arXiv

  61. [69]

    Robion C. Kirby. The topology of 4 -manifolds , volume 1374 of Lecture Notes in Mathematics . Springer-Verlag, Berlin, 1989

    1989

  62. [70]

    Singular R icci flows I

    Bruce Kleiner and John Lott. Singular R icci flows I . Acta Math. , 219(1):65--134, 2017

    2017

  63. [71]

    A uniqueness theorem for parabolic equations

    Fang-Hua Lin. A uniqueness theorem for parabolic equations. Comm. Pure Appl. Math. , 43(1):127--136, 1990

    1990

  64. [72]

    Closed mean curvature flows with asymptotically conical singularities

    Tang-Kai Lee and Xinrui Zhao. Closed mean curvature flows with asymptotically conical singularities. arXiv preprint arXiv:2405.15577 , 2024

    2024 arXiv

  65. [73]

    Lectures on the h -cobordism theorem

    John Milnor. Lectures on the h -cobordism theorem . Princeton University Press, Princeton, NJ, 1965. Notes by L. Siebenmann and J. Sondow

    1965

  66. [74]

    Ricci flow with surgery on three-manifolds

    Grisha Perelman. Ricci flow with surgery on three-manifolds. arXiv preprint math/0303109 , 2003

    2003 arXiv

  67. [75]

    Unique continuation for parabolic equations

    Chi-Cheung Poon. Unique continuation for parabolic equations. Comm. Partial Differential Equations , 21(3-4):521--539, 1996

    1996

  68. [76]

    On a class of minimal cones in R n

    Plinio Simoes. On a class of minimal cones in R n . Bull. Amer. Math. Soc. , 80:488--489, 1974

    1974

  69. [77]

    Singularity profile in the mean curvature flow

    Weimin Sheng and Xu-Jia Wang. Singularity profile in the mean curvature flow. Methods Appl. Anal. , 16(2):139--155, 2009

    2009

  70. [78]

    Multiplicity one for min-max theory in compact manifolds with boundary and its applications

    Ao Sun, Zhichao Wang, and Xin Zhou. Multiplicity one for min-max theory in compact manifolds with boundary and its applications. arXiv preprint arXiv:2011.04136 , 2020

    2011 arXiv

  71. [79]

    Initial perturbation of the mean curvature flow for asymptotical conical limit shrinker

    Ao Sun and Jinxin Xue. Initial perturbation of the mean curvature flow for asymptotical conical limit shrinker. arXiv preprint arXiv:2107.05066 , 2021

    2021 arXiv

  72. [80]

    Initial perturbation of the mean curvature flow for closed limit shrinker

    Ao Sun and Jinxin Xue. Initial perturbation of the mean curvature flow for closed limit shrinker. arXiv preprint arXiv:2104.03101 , 2021

    2021 arXiv

  73. [81]

    Generic mean curvature flows with cylindrical singularities

    Ao Sun and Jinxin Xue. Generic mean curvature flows with cylindrical singularities. arXiv preprint arXiv:2210.00419 , 2022

    2022 arXiv

  74. [82]

    Schoen and S

    R. Schoen and S. T. Yau. On the structure of manifolds with positive scalar curvature. Manuscripta Math. , 28(1-3):159--183, 1979

    1979

  75. [83]

    The topology of hypersurfaces moving by mean curvature

    Brian White. The topology of hypersurfaces moving by mean curvature. Communications in analysis and geometry , 3(2):317--333, 1995

    1995

  76. [84]

    Stratification of minimal surfaces, mean curvature flows, and harmonic maps

    Brian White. Stratification of minimal surfaces, mean curvature flows, and harmonic maps. Journal für die reine und angewandte Mathematik , 488:1--36, 1997

    1997

  77. [85]

    The size of the singular set in mean curvature flow of mean-convex sets

    Brian White. The size of the singular set in mean curvature flow of mean-convex sets. Journal of the American Mathematical Society , 13(3):665--695, 2000

    2000

  78. [86]

    The nature of singularities in mean curvature flow of mean-convex sets

    Brian White. The nature of singularities in mean curvature flow of mean-convex sets. Journal of the American Mathematical Society , 16(1):123--138, 2003

    2003

  79. [87]

    A local regularity theorem for mean curvature flow

    Brian White. A local regularity theorem for mean curvature flow. Annals of mathematics , pages 1487--1519, 2005

    2005

  80. [88]

    Currents and flat chains associated to varifolds, with an application to mean curvature flow

    Brian White. Currents and flat chains associated to varifolds, with an application to mean curvature flow. Duke Mathematical Journal , 148(1):41--62, 2009

    2009

  81. [89]

    The maximum principle for minimal varieties of arbitrary codimension

    Brian White. The maximum principle for minimal varieties of arbitrary codimension. Communications in Analysis and Geometry , 18(3):421--432, 2010

    2010

  82. [90]

    Topological change in mean convex mean curvature flow

    Brian White. Topological change in mean convex mean curvature flow. Inventiones mathematicae , 191(3):501--525, 2013

    2013

  83. [91]

    Subsequent singularities in mean-convex mean curvature flow

    Brian White. Subsequent singularities in mean-convex mean curvature flow. Calculus of Variations and Partial Differential Equations , 54(2):1457--1468, 2015

    2015

  84. [92]

    Mean curvature flow with boundary

    Brian White. Mean curvature flow with boundary. Ars Inven. Anal. , pages Paper No. 4, 43, 2021

    2021

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