The curve shortening flow with density of a spherical curve in codimension two

Francisco Vi\~nado-Lereu

REVIEW 2 major objections 4 minor 24 references

The curve shortening flow with density of a spherical curve in codimension two

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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

Pith's one-line read A spherical curve under mean curvature flow with density collapses in finite time to a round point or, if it runs forever, converges at infinity to a $\psi$-minimal curve or a point, according to the parabolicity or hyperbolicity of the…

desk verdict Smart reduction and a nice Gaussian classification, but the central Theorem 6 solves the opposite sign flow, so the qualitative theorems are not established. read the letter →

arxiv 1908.07441 v1 pith:AZ2PB3QH submitted 2019-08-20 math.DG

Francisco Viñado-Lereu This is my paper · ORCID

classification math.DG MSC 53C4435R01

keywords meancurvatureflowwithdensitycurveshorteningsphericalcurvescodimensiontworotationallysymmetricspacesmanifoldsparabolicityhyperbolicity

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

This paper studies the mean curvature flow with density for closed embedded curves that start on a geodesic sphere in a three-dimensional rotationally symmetric manifold with density $\xi=\phi(r)+\psi(\theta)$. It proves that every such flow splits exactly into a radial motion driven by the ordinary differential equation $R'(t)=-B(R(t))$ and an angular curve-shortening flow on the sphere, so the shape evolution is inherited from the known theory on surfaces. If the flow dies in finite time, the curve collapses to a point, away from the pole as a spherical round point; if it lives forever, the limit is controlled by the geometry at infinity. The main new result is a dichotomy: in a parabolic ambient space an unbounded flow subconverges to a closed $\psi$-minimal curve on the sphere at infinity, while in a hyperbolic space it either converges to a fixed curve at infinity or collapses to a point there. This transfers a classical dichotomy of the curve shortening problem to the codimension-two setting and links it to the potential theory of the ambient space.

What carries the argument

The load-bearing identity is the decomposition $\vec H_\xi=\vec k_{S_r,\psi}+\vec B(\gamma)$ (Lemma 1), which writes the ambient mean curvature vector with density as the geodesic curvature vector with density of the curve inside its geodesic sphere plus the radial vector $\vec B=-B(r)\partial_r$, where $B(r)=w'(r)/w(r)+\phi'(r)$. With this identity the solution is explicit: $\gamma(p,t)=\exp(\tilde\gamma(p,\tilde t(t)),(R(t)-r_0)\partial_r)$, where $R'(t)=-B(R(t))$ and $\tilde t(t)=\int_0^t(w(r_0)/w(R(t)))^2\,dt$; the problem reduces to the known $\psi$MCF on a sphere composed with a one-dimensional radial ODE. Whether the rescaled time $\tilde T$ is finite or infinite is controlled by the integral $-\operatorname{Area}(S_{r_0})\int_{r_0}^\infty 1/(B(r)\operatorname{Area}(S_r))\,dr$, and the standard characterization of parabolicity and hyperbolicity by the divergence of $\int^\infty 1/\operatorname{Area}(S_r)\,dr$ converts that integral into the theorem's dichotomy.

What would settle it

In the Gaussian case $\phi(r)=-\frac12\mu^2r^2$ with a non-geodesic initial curve, compute $\partial_s^n k_{\gamma(\cdot,t),S_{R(t)},\psi}$ as $t\to\infty$; any derivative that fails to converge uniformly to zero while the angular flow is still defined would refute equation (23) and the smooth subconvergence claim. Alternatively, check directly whether the estimate cited from page 23 of the earlier paper applies to spheres whose radius $R(t)$ changes with time, since that is the step the manuscript leaves unproved.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem C: for a spherical curve evolving by the mean curvature flow with density in $M^3_w$, if the solution exists for all time and is unbounded, its asymptotic behavior is decided by whether $M^3_w$ is parabolic or hyperbolic. In the parabolic case with $\liminf_{r\to\infty}B(r)$ finite, the flow topologically subconverges to $\gamma_\infty:S^1\to[0,\infty]\times S^2$, $p\mapsto(\infty,\chi(p))$, where $\chi$ is a closed embedded $\psi$-minimal curve. In the hyperbolic case with $\limsup_{r\to\infty}B(r)\neq0$, the flow either topologically converges to the curve $\gamma_\infty(p)=(\infty,\tilde\gamma(p,\tilde T))$ or to a single point on the sphere at infinity. The paper also proves that finite-time maximal solutions collapse to a point (Theorem A), that bounded eternal solutions $C^\infty$-subconverge to a $\psi$-minimal curve in a sphere where $B=0$ (Theorem B), and it works out the Euclidean Gaussian-density case in full, classifying each initial curve by the area it encloses.

Load-bearing premise

The argument's load-bearing premise is that a curvature-decay estimate imported from an earlier work—every derivative of the curve's density-weighted curvature inside its sphere tends uniformly to zero at infinity—remains valid when the sphere itself moves with the flow; the paper does not prove that transfer.

Editorial extensions

If this is right

Any finite-time singularity of a spherical solution is a collapse to a point: a spherical round point away from the pole, and at the pole (when $\phi$ has a $C^1$ extension) a blow-up that subconverges to a closed $\psi$-minimal curve.
A bounded eternal solution $C^\infty$-subconverges to a closed $\psi$-minimal curve contained in a $B$-minimal geodesic sphere, so such spheres are barriers for the flow.
In a parabolic ambient space, an unbounded eternal flow with $\liminf_{r\to\infty}B(r)$ finite forgets its angular shape and approaches a $\psi$-minimal curve on the sphere at infinity.
In a hyperbolic ambient space with $\limsup_{r\to\infty}B(r)\neq0$, the same flow either approaches a fixed curve on the sphere at infinity or collapses to a point there.
In Euclidean space with Gaussian density, the dichotomy becomes a computable area threshold: larger enclosed area sends the curve to a limit curve at infinity, the threshold area sends it to a point at infinity, and smaller area makes it collapse in finite time.

Reading between the lines

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

The same radial-angular splitting should extend to spherical submanifolds of any codimension in the same warped-product geometry, with the same radial ODE driven by $B$; the angular part would then be a higher-dimensional flow with density.
The Gaussian area threshold is a concrete place to test sharpness: numerical evolutions just above and just below the threshold should show convergence to a curve versus a point at infinity, while behavior exactly at the threshold would expose whether the missing curvature estimate is needed.
If the imported estimate (23) cannot be transferred to the moving spheres, the topological part of Theorem C may survive, but the smooth subconvergence would require a new argument controlling derivatives along the time-dependent family of spheres.
Read through potential theory, the dichotomy says a recurrent ('parabolic') ambient space forces the angular flow to last forever and settle down, while a transient ('hyperbolic') ambient space lets the angular flow finish in finite rescaled time before the curve reaches infinity.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

Fernando — this one is a mixed bag. The core idea is genuinely nice: decompose the ξMCF of a spherical curve into a radial ODE driven by B(r) and the ψMCF on a fixed sphere, with the time change (17). That reduction, and the parabolic/hyperbolic dichotomy for unbounded solutions, is a real contribution, and the Gaussian density classification with the explicit area threshold is the right kind of payoff. If the signs in Theorem 6 were fixed, this would be a solid subfield paper.

The problem is that they aren't. Problem (6) is stated as ∂γ/∂t = −H_ξ. Lemma 1 gives H_ξ = k_{S_r,ψ} + B_vec, with B_vec = −B ∂_r. So −H_ξ = −k_{S_r,ψ} + B ∂_r. That means the radial component of the stated velocity is +B ∂_r, so R' should be +B(R). Theorem 6 instead takes R' = −B, and the proof computes ∂γ/∂t = B_vec + k_{S_R,ψ} = H_ξ. In other words, the solution constructed in Theorem 6 solves ∂γ/∂t = +H_ξ, not problem (6). This is not a harmless convention slip: every theorem that says 'unbounded solution requires B < 0 at infinity' (Theorems B, C, 11) uses the sign from the equation that is actually solved, not the stated one. If problem (6) were solved with the minus sign, R would grow exactly when B > 0, and the whole dichotomy would flip. The paper never acknowledges this.

There is a second, independent gap: the C^∞ subconvergence claims in Theorems 10(ii) and 11(i) rest on the uniform derivative estimate (23), attributed to 'Step 4 on page 23 of [15]' and equation (23). That estimate is not stated or proved in this paper, and it concerns curvature on a fixed sphere, while the sphere S_{R(t)} moves with time. Transferring it takes argument that isn't there. So even if the sign were fixed, those convergence claims are not established.

What should you take from this? The paper is worth reading for the reduction and the Gaussian example, but the main theorems as stated do not hold up. The sign error is fixable — it may just be a matter of changing the convention in (6) or in the ODE — but the author needs to present a consistent formulation and rework the estimates. I'd send it to a referee: the idea is good enough that a serious referee could help turn it into a correct paper. But I wouldn't cite it as is.

Referee Report

2 major / 4 minor

Summary. The paper studies the curve shortening flow with density for spherical curves in a 3-dimensional rotationally symmetric space with density (M_w^3, g_w, ξ), where ξ splits into a radial part φ and an angular part ψ. The author constructs the solution by reducing it to the curve shortening flow with density on a fixed geodesic sphere and a scalar radial ODE, and then analyzes finite-time collapse and infinite-time behavior. The main classification results, Theorems A–C, assert that finite-time solutions collapse to a round point or the pole, and that unbounded infinite-time solutions converge topologically to a ψ-minimal curve or a point at infinity, depending on the parabolicity or hyperbolicity of M_w^3 and on the limiting behavior of the function B(r) = w'(r)/w(r) + φ'(r). A final section applies the theory to the Gaussian density in R^3.

Significance. If the main theorems were correct, the paper would provide a clean reduction of a codimension-two curve flow with density to a surface flow, together with a potential-theoretic dichotomy for spherical curves and explicit Gaussian examples. The manuscript has genuine strengths: the reduction is explicit, the comparison principle is used in a natural way, and the Gaussian example is worked out in detail. However, the central existence proof contains a sign error that invalidates the stated problem and the theorems derived from it. The paper also relies on an unproved, imported derivative estimate for the subconvergence claims. As it stands, the central claims do not follow from the arguments given.

major comments (2)

[Theorem 6; Eq. (6)-(16)] Problem (6) prescribes ∂γ/∂t = −Hξ. By Lemma 1, Hξ = k_{S_R,ψ} + B, with B = −B(r)∂r, so the prescribed radial velocity is +B(r)∂r. In the proof of Theorem 6 the constructed solution satisfies ∂γ/∂t = R'∂r + (w(r0)/w(R))^2 exp_*(−k_{S_r0,ψ}) = −B(R)∂r + k_{S_R,ψ} = B + k_{S_R,ψ} = Hξ, using Lemma 3 and Lemma 1. Thus the construction solves ∂γ/∂t = +Hξ, not the stated problem (6). The discrepancy is visible in the ODE (16): the text sets R' = −B, which is the radial component of +Hξ; solving (6) would require R' = +B. This is load-bearing: Theorem 11(ii) and Theorem C derive 'B(r) < 0 for all r ≥ r0' as the necessary condition for unboundedness from R' = −B, whereas the stated problem would require B > 0; Section 6's Gaussian ODE (30) also follows the R' = −B convention. A reparametrization of t cannot reverse a one-sided radial drift, so this is not a cosmetic sign error.
[Section 5, display (23)] The claim that all derivatives of the weighted geodesic curvature of the angular flow converge uniformly to zero as t goes to infinity is asserted by reference to 'Step 4 on page 23 of [15]' together with Lemma 4, but it is not proved in this paper. The setting of [15] is a flow in a fixed surface, whereas here the sphere S_{R(t)} changes with time; transferring the estimate requires argumentation that is not provided. This estimate is used to bound |∂^n_α γ̂| in (28) and hence to obtain the C^∞ subconvergence in Theorem 11(i). Without it the subconvergence conclusion is unsupported. If the conclusion can be obtained more directly from Theorem 3 applied to the fixed-sphere flow ~γ, the proof should say so explicitly.

minor comments (4)

[Equation (21)] The summation notation '∑_{n,1}^{1,n−1}' is not explained and appears malformed; the multi-index sum over i, J, K should be defined precisely in the text.
[Proposition 2] The formula for the derivative of the inverse function is incorrect as printed: '(L|_{[r0,∞)})^{-1}(r) = 1/L|_{[r0,∞)}(r) = 1/B(r)' should involve L' composed with L^{-1}, not L itself. The equivalence may be true, but the proof needs to be rewritten.
[Lemma 5 and Eq. (24)] In the computation of ∂^3_s γ and ∂^4_s γ, the term ∂_s(w'/w) vanishes because w'/w is constant along a spherical curve at fixed R(t); noting this explicitly would clarify the polynomial expressions f_n, g_n, h_n in (24).
[Introduction, after Eq. (11)] The heuristic discussion of the integral (10) is useful but imprecise: the notation '~' in (11) is not defined, and the precise role of parabolicity/hyperbolicity is only made clear much later in Theorem 11.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central reduction is constructive and the cited prior-work results are independent support, not self-referential inputs.

full rationale

No circularity found. Theorem 6 reduces the ξMCF (6) to the ψMCF (7) plus a scalar ODE via a genuinely constructive separation-of-variables argument (Lemmas 1–3), not by assuming the conclusion. The later convergence conclusions invoke Theorem 3 and Step 4 of [15]; these are prior published results with stated hypotheses (compact domain, conditions (13)–(14)) that do not include the target theorem, so under the rules they are independent support rather than circular self-citation. The only serious defect is a sign inconsistency in the proof of Theorem 6: the computed time derivative equals +Hξ, whereas problem (6) prescribes −Hξ; this is a correctness/rigor issue, not a circularity. The C∞ estimate (23) is borrowed rather than reproved, another support gap, not circularity. No fitted parameter is relabeled as a prediction and no known result is merely renamed.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new free parameters or entities. The central claim rests on several imported results: the convergence theory of the ψMCF from the author's prior work [15], the uniqueness theory for MCF from Hamilton/Smoczyk, and the potential-theoretic characterization of parabolic/hyperbolic manifolds. The most fragile import is the unstated Step 4 of [15] used for the derivative estimate (23).

assumptions (5)

domain assumption Theorem 3 of [15]: on a compact surface with density satisfying (13)-(14), the ψMCF subconverges C^m to a closed ψ-minimal curve
Used to conclude convergence of the angular flow in Theorems 10 and 11. Not proved in the present paper.
domain assumption Step 4 on page 23 of [15]: uniform convergence to zero of all derivatives of the weighted geodesic curvature along the ψMCF
Basis for (23), the key derivative estimate in Theorem 11(i). Not stated or proved here.
standard math Hamilton's short-time existence and uniqueness theorem for MCF in higher codimension, as surveyed by Smoczyk [21]
Used in Theorem 6 to justify uniqueness of the solution to (6).
standard math Characterization of parabolicity/hyperbolicity of rotationally symmetric manifolds via convergence/divergence of ∫ dr/Area(S_r) (Grigor'yan [11])
Used in Theorem 11 to relate finiteness of the angular time parameter to the manifold type.
domain assumption The area variation formula dA/d~t = -2π + A/r0² for the curve shortening flow on S² under the chosen orientation
Used in Section 6 to compute the maximal time of the angular flow; its sign depends on the same convention as the flow equation.

how reviews work

0 comments

Cite this review

Pith. "Pith review of The curve shortening flow with density of a spherical curve in codimension two." pith.science (2026). https://pith.science/paper/AZ2PB3QH

@misc{pith2026190807441,
  author       = {Pith},
  title        = {Pith review of: The curve shortening flow with density of a spherical curve in codimension two},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AZ2PB3QH}},
  note         = {Machine review of arXiv:1908.07441}
}

abstract

In the present paper we carry out a systematic study about the flow of a spherical curve by the mean curvature flow with density in a 3-dimensional rotationally symmetric space with density $(M^3_w,\:g_w,\:\xi)$ where the density $\xi$ decomposes as sum of a radial part $\varphi$ and an angular part $\psi$. We analyse how either the parabolicity or the hyperbolicity of $(M^3_w,\:g_w)$ condition the behaviour of the flow when the solution goes to infinity.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages

[15]

The curve shortening problem associated to a density

Miquel, V., and Vi ˜nado-Lereu, F. The curve shortening problem associated to a density. Calculus of Variations and Partial Diﬀerential Equations 55:61 , 3 (June 2016), 1 – 30

work page 2016
[1]

Parabolic equations for curves on surfaces

Angenent, S. Parabolic equations for curves on surfaces. i. curves with p-integrable curvature. Annals of Mathematics (2) 132 , 3 (1990), 451–483

work page 1990
[2]

Parabolic equations for curves on surfaces

Angenent, S. Parabolic equations for curves on surfaces. ii. intersec- tions, blow-up and generalized solutions. Annals of Mathematics (2) 133, 1 (January 1991), 171–215

work page 1991
[3]

Gaussian mean curvature ﬂow

Borisenko, A., and Miquel, V. Gaussian mean curvature ﬂow. Journal of Evolution Equations 10 (2010), 415–423

work page 2010
[4]

M., and Miquel, V

Castro, I., Lerma, A. M., and Miquel, V. Evolution by mean curvature ﬂow of lagrangian spherical surfaces in complex e uclidean plane. Journal of Mathematical Analysis and Applications 462 (2018), 637–647

work page 2018
[5]

Mean curvature ﬂow of surface in 4-manifolds

Chen, J., and Li, J. Mean curvature ﬂow of surface in 4-manifolds. Advances in Mathematics 163 , 2 (November 2001), 287–309

work page 2001
[6]

The Curve Shortening Problem

Chou, K.-S., and Zhu, X.-P. The Curve Shortening Problem . Chap- man and Hall/CRC, Boca Raton, FL, March 2001

work page 2001
[7]

The heat equation shrinking convex plane curves

Gage, M., and Hamilton, R. The heat equation shrinking convex plane curves. Journal Diﬀerential Geometry 23 (1986), 69–96

work page 1986

Show all 24 references

[8]

Gage, M. E. Curve shortening makes convex curves circular. Inven- tiones mathematicae 76 (1984), 357–364

1984
[9]

Gage, M. E. Curve shortening on surfaces. Annales Scientiﬁques de l’´Ecole Normale Sup´ erieure 23, 2 (1990), 229–256

1990
[10]

Shortening embedded curves

Grayson, M. Shortening embedded curves. Annals of Mathematics 129 (1989), 71–111. 30

1989
[11]

Analytic and geometric background of recurrence and non-explosion of the brownian motion on riemannian manifol ds

Grigor’yan, A. Analytic and geometric background of recurrence and non-explosion of the brownian motion on riemannian manifol ds. Bull. Amer. Math. Soc. (N.S.) 36 (1999), 135–249

1999
[12]

Three-manifolds with positive ricci curvature

Hamilton, R. Three-manifolds with positive ricci curvature. Journal Diﬀerential Geometry 17 , 2 (1982), 255–306

1982
[13]

Asymptotic behavior for singularities of the mean curva- ture ﬂow

Huisken, G. Asymptotic behavior for singularities of the mean curva- ture ﬂow. Journal Diﬀerential Geometry 31 , 1 (1990), 285 – 299

1990
[14]

Curve shortening in a Riemannian manifold , vol

Ma, L., and Chen, D. Curve shortening in a Riemannian manifold , vol. 186. Springer-Verlag, October 2007

2007
[16]

Type i singularities in the curve shortening ﬂow associated to a density

Miquel, V., and Vi ˜nado-Lereu, F. Type i singularities in the curve shortening ﬂow associated to a density. The Journal of Geometric Analysis 28 , 3 (July 2018), 2361–2394

2018
[17]

Oaks, J. A. Singularities and self intersections of curve evolving on surfaces. Indiana University Mathematics Journal 43 , 3 (1994), 959– 981

1994
[18]

Semi-Riemannian Geometry with Applications to Rela- tivity

O’Neill, B. Semi-Riemannian Geometry with Applications to Rela- tivity. 1983

1983
[19]

C., and Smoczyk, K

Schn¨urer, O. C., and Smoczyk, K. Evolution of hypersurfaces in central force ﬁelds. Journal f¨ ur die reine und angewandte Mathematik 550 (2002), 77–95

2002
[20]

Symmetric hypersurfaces in riemannian manifolds con- tracting to lie-groups by their mean curvature

Smoczyk, K. Symmetric hypersurfaces in riemannian manifolds con- tracting to lie-groups by their mean curvature. Calculus of Variations and Partial Diﬀerential Equations 4 , 2 (February 1996), 155 – 170

1996
[21]

Mean curvature ﬂow in higher codimension: Introduc- tion and survey

Smoczyk, K. Mean curvature ﬂow in higher codimension: Introduc- tion and survey. In Global Diﬀerential Geometry. , M. S. Christian B¨ ar, Joachim Lohkamp, Ed., vol. 17 of Springer Proceedings in Mathematics. Springer, Berlin, Heidelberg, 2012, pp. 231–274

2012
[22]

Mean curvature ﬂow of surfaces in einstein four- manifolds

W ang, M.-T. Mean curvature ﬂow of surfaces in einstein four- manifolds. Journal of Diﬀerential Geometry 57 , 2 (2001), 301–338

2001
[23]

Curve shortening ﬂows in warped product manifolds

Zhou, H. Curve shortening ﬂows in warped product manifolds. Pro- ceedings of the American Mathematical Society 145 , 10 (October 2017), 4503–4516. 31

2017
[24]

Asymptotic behavior of anisotropic curve ﬂows

Zhu, X.-P. Asymptotic behavior of anisotropic curve ﬂows. Journal Diﬀerential Geometry 48 (1998), 225–274. Department of Mathematics, School of Technology and Experi mental Sci- ences, Universitat Jaume I, Av. de Vicente Sos Baynat, s/n, E -12071 Castell´ o de la Plana, Spain ...

1998

Pith tools

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

[1] [15]

The curve shortening problem associated to a density

Miquel, V., and Vi ˜nado-Lereu, F. The curve shortening problem associated to a density. Calculus of Variations and Partial Diﬀerential Equations 55:61 , 3 (June 2016), 1 – 30

work page 2016

[2] [1]

Parabolic equations for curves on surfaces

Angenent, S. Parabolic equations for curves on surfaces. i. curves with p-integrable curvature. Annals of Mathematics (2) 132 , 3 (1990), 451–483

work page 1990

[3] [2]

Parabolic equations for curves on surfaces

Angenent, S. Parabolic equations for curves on surfaces. ii. intersec- tions, blow-up and generalized solutions. Annals of Mathematics (2) 133, 1 (January 1991), 171–215

work page 1991

[4] [3]

Gaussian mean curvature ﬂow

Borisenko, A., and Miquel, V. Gaussian mean curvature ﬂow. Journal of Evolution Equations 10 (2010), 415–423

work page 2010

[5] [4]

M., and Miquel, V

Castro, I., Lerma, A. M., and Miquel, V. Evolution by mean curvature ﬂow of lagrangian spherical surfaces in complex e uclidean plane. Journal of Mathematical Analysis and Applications 462 (2018), 637–647

work page 2018

[6] [5]

Mean curvature ﬂow of surface in 4-manifolds

Chen, J., and Li, J. Mean curvature ﬂow of surface in 4-manifolds. Advances in Mathematics 163 , 2 (November 2001), 287–309

work page 2001

[7] [6]

The Curve Shortening Problem

Chou, K.-S., and Zhu, X.-P. The Curve Shortening Problem . Chap- man and Hall/CRC, Boca Raton, FL, March 2001

work page 2001

[8] [7]

The heat equation shrinking convex plane curves

Gage, M., and Hamilton, R. The heat equation shrinking convex plane curves. Journal Diﬀerential Geometry 23 (1986), 69–96

work page 1986