Propagation of a Mean Curvature Flow in a Cone

Bendong Lou

arxiv: 1907.11520 · v1 · pith:E4I4OTZVnew · submitted 2019-07-26 · 🧮 math.DG

Propagation of a Mean Curvature Flow in a Cone

Bendong Lou This is my paper

Pith reviewed 2026-05-24 15:29 UTC · model grok-4.3

classification 🧮 math.DG

keywords mean curvature flowconeself-similar solutionshomogenizationcontact angleradial symmetryglobal existenceperiodic condition

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The pith

A family of self-similar solutions yields global existence for radially symmetric mean curvature flows in a cone with ε-periodic contact angle and characterizes the homogenization limit as ε vanishes.

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

The paper constructs a family of self-similar solutions to obtain a priori estimates on radially symmetric solutions of mean curvature flow inside a cone subject to an ε-periodic contact angle condition, which in turn proves that these solutions exist for all time. It next passes to the homogenization limit ε to 0 and shows that the evolving surface stays within an error of order ε to the power 1/6 of the slowest member of the family over any finite time interval. The construction therefore controls both the long-time behavior for positive ε and the limiting profile when the period shrinks to zero.

Core claim

By constructing a family of self-similar solutions, a priori estimates are derived for the radially symmetric solutions of the mean curvature flow in a cone with ε-periodic contact angle, proving their global existence in time. In the homogenization limit as ε tends to zero, the solution is characterized by the slowest self-similar solution with an error of O(ε^{1/6}) over finite time intervals.

What carries the argument

The family of self-similar solutions, which supplies both the uniform estimates needed for global existence at fixed ε and the limiting profile used in the homogenization argument.

If this is right

Radially symmetric solutions exist globally in time for every fixed positive ε.
The evolving hypersurface remains controlled by the constructed self-similar profiles for all time.
In the limit ε to 0 the solution converges to the slowest self-similar solution at rate O(ε^{1/6}) on finite time intervals.
The homogenization description holds uniformly on any compact time interval.

Where Pith is reading between the lines

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

The same self-similar family might serve as barriers for non-radially symmetric solutions if suitable comparison principles can be established.
The ε^{1/6} rate may be sharp or improvable by a more refined asymptotic analysis near the slowest profile.
Numerical schemes could approximate the flow by solving the ODE that defines the self-similar solutions and then evolving the initial data toward them.

Load-bearing premise

A family of self-similar solutions exists that supplies a priori estimates strong enough to prove global existence for every radially symmetric solution obeying the ε-periodic contact angle condition.

What would settle it

A radially symmetric solution satisfying the ε-periodic contact angle condition that develops a singularity in finite time, or a sequence of solutions whose deviation from the slowest self-similar profile exceeds order ε^{1/6} in the small-ε limit.

read the original abstract

We consider a mean curvature flow in a cone, that is, a hypersurface in a cone which moves toward the opening with normal velocity equaling to the mean curvature, and the contact angle between the hypersurface and the cone boundary being $\varepsilon$-periodic in its position. First, by constructing a family of self-similar solutions, we give a priori estimates for the radially symmetric solutions and prove the global existence. Then we consider the homogenization limit as $\ve\to 0$, and use {\it the slowest self-similar solution} to characterize the solution, with error $O(1)\ve^{1/6}$, in some finite time interval.

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 global existence and homogenization claims both hinge on whether the self-similar family actually supplies uniform a priori estimates, which the abstract asserts but does not demonstrate.

read the letter

The paper claims global existence for radially symmetric mean curvature flow in a cone under an ε-periodic contact angle condition, obtained by constructing a family of self-similar solutions that deliver a priori estimates, followed by a homogenization limit as ε→0 that picks out the slowest member of that family to get an O(ε^{1/6}) characterization over a finite time interval. That combination of self-similar bounds plus a quantitative rate looks like the concrete new piece relative to standard MCF work in conical domains. The approach is direct and the error rate is explicit, which is better than many qualitative homogenization statements. The construction is presented as independent of the target solution, so no obvious circularity. The result is new for this periodic-contact-angle setup in a cone. The load-bearing step is exactly the one the stress-test flags: whether the self-similar family exists across the admissible initial data and actually produces ε-independent bounds strong enough for global existence. The abstract states this without derivation details, function-space statements, or identification of the slowest solution, so the claim cannot be checked from the given text. If the bounds turn out to be only ε-dependent or fail for some data, both existence and the error estimate fall. No other part of the argument compensates. This is narrow-scope work aimed at specialists in geometric PDEs who already know the standard MCF literature and are interested in periodic boundary conditions or conical obstacles. A reader looking for explicit rates in homogenization of flows might extract something usable if the estimates hold. It deserves a serious referee to check the self-similar construction and the error derivation rather than a desk reject.

Referee Report

2 major / 2 minor

Summary. The manuscript considers mean curvature flow of hypersurfaces inside a cone with an ε-periodic contact angle boundary condition. By constructing a family of self-similar solutions, it derives a priori estimates for radially symmetric solutions and proves global existence. It then studies the homogenization limit ε→0 and characterizes the limiting solution via the slowest self-similar solution, obtaining an error bound of O(1)ε^{1/6} on a finite time interval.

Significance. If the asserted family of self-similar solutions supplies the claimed ε-independent a priori estimates, the work would supply a new technique for global existence and homogenization under rapidly oscillating boundary conditions in geometric flows. The explicit (though non-sharp) error rate in the limit is a concrete contribution that could be useful in related problems.

major comments (2)

[self-similar solutions construction] The central claim of global existence for all radially symmetric solutions rests on the construction of a family of self-similar solutions that furnish ε-independent a priori estimates (abstract and the section describing the construction). The manuscript must explicitly verify that these bounds remain uniform in ε for the full range of admissible initial data satisfying the periodic contact-angle condition; without this verification the global-existence statement and the subsequent homogenization limit both fail.
[homogenization limit] In the homogenization analysis, the selection of the slowest self-similar solution to obtain the O(1)ε^{1/6} characterization requires a precise statement of the function spaces in which the estimates and the limit hold, together with a justification that this particular member of the family controls the solution (abstract). The current presentation does not indicate these spaces or the comparison argument.

minor comments (2)

The notation for the cone, the radial symmetry assumption, and the precise form of the ε-periodic contact angle condition would benefit from an introductory figure or diagram.
A short comparison paragraph relating the constructed self-similar solutions to existing literature on self-similar mean curvature flows would improve context.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the two major comments point by point below. Both concerns can be resolved by adding explicit verifications and clarifications in a revised manuscript.

read point-by-point responses

Referee: [self-similar solutions construction] The central claim of global existence for all radially symmetric solutions rests on the construction of a family of self-similar solutions that furnish ε-independent a priori estimates (abstract and the section describing the construction). The manuscript must explicitly verify that these bounds remain uniform in ε for the full range of admissible initial data satisfying the periodic contact-angle condition; without this verification the global-existence statement and the subsequent homogenization limit both fail.

Authors: We agree that the ε-uniformity of the a priori estimates must be verified explicitly for the full range of admissible initial data. In the revised manuscript we will add a subsection immediately following the construction of the self-similar solutions that proves the estimates are independent of ε. The argument will use the radial symmetry together with the periodicity of the contact-angle condition to show that the comparison barriers remain valid uniformly for all admissible initial data; this will directly support both the global-existence theorem and the subsequent homogenization analysis. revision: yes
Referee: [homogenization limit] In the homogenization analysis, the selection of the slowest self-similar solution to obtain the O(1)ε^{1/6} characterization requires a precise statement of the function spaces in which the estimates and the limit hold, together with a justification that this particular member of the family controls the solution (abstract). The current presentation does not indicate these spaces or the comparison argument.

Authors: We acknowledge that the function spaces and the comparison argument justifying the choice of the slowest self-similar solution were not stated with sufficient precision. In the revision we will (i) specify the precise function spaces (radial graphs in C^{2,α} on compact time intervals, with the contact-angle condition incorporated) in which the O(ε^{1/6}) error estimate holds, and (ii) supply a short comparison-principle argument showing that any solution starting from admissible data is trapped between the slowest and fastest members of the family, so that the slowest member furnishes the sharp upper envelope in the homogenization limit. These additions will be placed in the homogenization section and referenced from the abstract. revision: yes

Circularity Check

0 steps flagged

No circularity: self-similar family constructed independently to supply estimates

full rationale

The paper states it constructs a family of self-similar solutions directly from the mean curvature flow equation in the cone, then applies those solutions to obtain a priori estimates that prove global existence for radially symmetric solutions with the ε-periodic contact angle condition. The homogenization limit then selects the slowest member of this independently constructed family. No quoted step reduces a claimed result to a definition of itself, a fitted parameter renamed as a prediction, or a self-citation chain; the derivation chain remains self-contained against the flow PDE and the constructed family.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper introduces no free parameters, no new invented entities, and relies only on standard background results from differential geometry and parabolic PDE theory.

axioms (1)

standard math Standard existence, regularity, and comparison principles for mean curvature flow in smooth domains.
Invoked implicitly to justify a priori estimates and passage to the homogenization limit.

pith-pipeline@v0.9.0 · 5624 in / 1283 out tokens · 41534 ms · 2026-05-24T15:29:41.088441+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

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work page 1990
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work page 1991
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Altschuler and L

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Altschuler and L

S. Altschuler and L. Wu, Translating surfaces of the non-parametric mean curvature ﬂow with prescribed contact angle , Calc. Var. Partial Diﬀerential Equations, 2 (1994), 101-1 11

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Cai and B

J. Cai and B. Lou, Convergence in a quasilinear parabolic equation with Neuma nn boundary conditions , Nonlinear Anal., 74 (2011), 1426-1435

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Chang, J.-S

Y.-L. Chang, J.-S. Guo and Y. Kohsaka, On a two-point free boundary problem for a quasilinear parab olic equation, Asymptotic Anal., 34 (2003), 333-358

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Chen and J.-S

X. Chen and J.-S. Guo, Motion by curvature of planar curves with end points moving f reely on a line , Math. Ann., 350 (2011), 277-311

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Chou and X.P

K.S. Chou and X.P. Zhu, The Curve Shortening Problem, Cha pman & all/CRC, New York, 2001

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M. Gage and R. Hamilton, The heat equation shrinking convex plane curves , J. Diﬀerential Geom., 23 (1986), 69-96

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Grayson, Shortening embedded curves , Ann

M. Grayson, Shortening embedded curves , Ann. of Math., (2) 129 (1989), 71-111

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M. Grayson, The heat equation shrinks embedded plane curves to round poi nts, J. Diﬀerential Geom., 26 (1987), 285-314

work page 1987
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J.-S. Guo, H. Matano, M. Shimojo and C.-H Wu, On a free boundary problem for the curvature ﬂow with driving force , Arch. Ration. Mech. Anal., 219 (2016), 1207-1272

work page 2016
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Guo and B

J.-S. Guo and B. Hu, On a two-point free boundary problem , Quart. Appl. Math., 64 (2006), 413-431

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

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

work page 1984
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Huisken, Asymptotic behavior for singularities of the mean curvatur e ﬂow , J

G. Huisken, Asymptotic behavior for singularities of the mean curvatur e ﬂow , J. Diﬀerential Geom., 31 (1990), 285-299

work page 1990
[16]

Kohsaka, Free boundary problem for quasilinear parabolic equation w ith ﬁxed angle of contact to a boundary, Nonlinear Anal., 45 (2001), 865-894

Y. Kohsaka, Free boundary problem for quasilinear parabolic equation w ith ﬁxed angle of contact to a boundary, Nonlinear Anal., 45 (2001), 865-894

work page 2001
[17]

Kohsaka and B

Y. Kohsaka and B. Lou, Curvature ﬂow in a sector with undulating boundaries , in preparation

work page
[18]

G. M. Lieberman, Second Order Parabolic Diﬀerential Eq uations, World Scientiﬁc, 1996

work page 1996
[19]

B. Lou, H. Matano and K. Nakamura, Recurrent traveling waves in a two-dimensional saw-toothe d cylinder and their average speed , J. Diﬀerential Equations, 255 (2013), 3357-3411

work page 2013
[20]

Matano, K.I

H. Matano, K.I. Nakamura and B. Lou, Periodic traveling waves in a two-dimensional cylinder wit h saw-toothed boundary and their homogenization limit , Netw. Heterog. Media, 1 (2006) 537-568

work page 2006
[21]

N. Hamamuki, Asymptotically self-similar solutions to curvature ﬂow eq uations with prescribed contact angle and their applications to groove proﬁles due to evapor ation-condensation, Adv. Diﬀerential Equa- tions, 19 (2014), 317-358

work page 2014
[22]

Yuan and B

L. Yuan and B. Lou, Entire solutions of a mean curvature ﬂow connecting two peri odic traveling waves , Appl. Math. Letters, 87 (2019), 73-79

work page 2019

[1] [1]

Angenent, Parabolic equations for curves on surfaces

S. Angenent, Parabolic equations for curves on surfaces. I. Curves with p -integrable curvature, Ann. of Math., (2) 132 (1990), 451-483

work page 1990

[2] [2]

Angenent, Parabolic equations for curves on surfaces

S. Angenent, Parabolic equations for curves on surfaces. II. Intersecti ons, blow-up and generalized solu- tions, Ann. of Math., (2) 133 (1991), 171-215

work page 1991

[3] [3]

Altschuler and L

S. Altschuler and L. Wu, Convergence to translating solitons for a class of quasilin ear parabolic equations with ﬁxed angle of contact to a boundary , Math. Ann., 295 (1993), 761–765

work page 1993

[4] [4]

Altschuler and L

S. Altschuler and L. Wu, Translating surfaces of the non-parametric mean curvature ﬂow with prescribed contact angle , Calc. Var. Partial Diﬀerential Equations, 2 (1994), 101-1 11

work page 1994

[5] [5]

Cai and B

J. Cai and B. Lou, Convergence in a quasilinear parabolic equation with Neuma nn boundary conditions , Nonlinear Anal., 74 (2011), 1426-1435

work page 2011

[6] [6]

Chang, J.-S

Y.-L. Chang, J.-S. Guo and Y. Kohsaka, On a two-point free boundary problem for a quasilinear parab olic equation, Asymptotic Anal., 34 (2003), 333-358

work page 2003

[7] [7]

Chen and J.-S

X. Chen and J.-S. Guo, Motion by curvature of planar curves with end points moving f reely on a line , Math. Ann., 350 (2011), 277-311

work page 2011

[8] [8]

Chou and X.P

K.S. Chou and X.P. Zhu, The Curve Shortening Problem, Cha pman & all/CRC, New York, 2001

work page 2001

[9] [9]

Gage and R

M. Gage and R. Hamilton, The heat equation shrinking convex plane curves , J. Diﬀerential Geom., 23 (1986), 69-96

work page 1986

[10] [10]

Grayson, Shortening embedded curves , Ann

M. Grayson, Shortening embedded curves , Ann. of Math., (2) 129 (1989), 71-111

work page 1989

[11] [11]

Grayson, The heat equation shrinks embedded plane curves to round poi nts, J

M. Grayson, The heat equation shrinks embedded plane curves to round poi nts, J. Diﬀerential Geom., 26 (1987), 285-314

work page 1987

[12] [12]

J.-S. Guo, H. Matano, M. Shimojo and C.-H Wu, On a free boundary problem for the curvature ﬂow with driving force , Arch. Ration. Mech. Anal., 219 (2016), 1207-1272

work page 2016

[13] [13]

Guo and B

J.-S. Guo and B. Hu, On a two-point free boundary problem , Quart. Appl. Math., 64 (2006), 413-431

work page 2006

[14] [14]

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

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

work page 1984

[15] [15]

Huisken, Asymptotic behavior for singularities of the mean curvatur e ﬂow , J

G. Huisken, Asymptotic behavior for singularities of the mean curvatur e ﬂow , J. Diﬀerential Geom., 31 (1990), 285-299

work page 1990

[16] [16]

Kohsaka, Free boundary problem for quasilinear parabolic equation w ith ﬁxed angle of contact to a boundary, Nonlinear Anal., 45 (2001), 865-894

Y. Kohsaka, Free boundary problem for quasilinear parabolic equation w ith ﬁxed angle of contact to a boundary, Nonlinear Anal., 45 (2001), 865-894

work page 2001

[17] [17]

Kohsaka and B

Y. Kohsaka and B. Lou, Curvature ﬂow in a sector with undulating boundaries , in preparation

work page

[18] [18]

G. M. Lieberman, Second Order Parabolic Diﬀerential Eq uations, World Scientiﬁc, 1996

work page 1996

[19] [19]

B. Lou, H. Matano and K. Nakamura, Recurrent traveling waves in a two-dimensional saw-toothe d cylinder and their average speed , J. Diﬀerential Equations, 255 (2013), 3357-3411

work page 2013

[20] [20]

Matano, K.I

H. Matano, K.I. Nakamura and B. Lou, Periodic traveling waves in a two-dimensional cylinder wit h saw-toothed boundary and their homogenization limit , Netw. Heterog. Media, 1 (2006) 537-568

work page 2006

[21] [21]

N. Hamamuki, Asymptotically self-similar solutions to curvature ﬂow eq uations with prescribed contact angle and their applications to groove proﬁles due to evapor ation-condensation, Adv. Diﬀerential Equa- tions, 19 (2014), 317-358

work page 2014

[22] [22]

Yuan and B

L. Yuan and B. Lou, Entire solutions of a mean curvature ﬂow connecting two peri odic traveling waves , Appl. Math. Letters, 87 (2019), 73-79

work page 2019