Uniqueness of self-similar solutions to flows by quotient curvatures

Li Chen; Shanze Gao

REVIEW 2 major objections 5 minor 17 references

Uniqueness of self-similar solutions to flows by quotient curvatures

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper proves that every closed strictly convex self-similar shrinker under a quotient of elementary-symmetric curvature functions, with exponent above 1/(k−l), is a round sphere.

desk verdict A clean, genuine extension of the self-similar uniqueness results to quotient curvature flows, but the proof's key algebraic estimate is imported from a self-cited preprint without proof, so the paper is conditionally correct rather than fully self-contained. read the letter →

arxiv 1908.04699 v1 pith:P2Y6WCOK submitted 2019-08-13 math.DG

Li Chen , Shanze Gao This is my paper · ORCID

classification math.DG MSC 35J1535J6053C44

keywords uniquenessconvexhypersurfacesquotientcurvaturesself-similarsolutionssupportfunctionmaximumprincipleelementarysymmetricpolynomialscurvatureflow

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 aims to close a gap in the uniqueness theory of self-similar shrinkers: when a closed strictly convex hypersurface moves by a quotient of elementary-symmetric curvature functions, the only self-similar solutions should be round spheres. The authors prove that a hypersurface satisfying $(\sigma_k(\kappa)/\sigma_l(\kappa))^\alpha = \langle X, \nu\rangle$ with $0\le l1/(k-l)$ must be a standard sphere. This matters because self-similar solutions describe the singularity profiles of curvature flows, and uniqueness says the profile is always spherical in this family. The proof works in support-function form: for $1\le l

What carries the argument

The central objects are the two comparison functions $W$ and $Z$. $W$ is the function $u\lambda_{\max}(b_{ij})-\frac{\beta}{2}(u^2+|Du|^2)$; its maximum points are controlled by Lemma 3.1, a modification of a lemma from [8]. $Z$ is the function $uG-\frac{n\beta}{2}(u^2+|Du|^2)$ with $G=\frac{n}{k}(\sigma_1-\frac{(k+1)\sigma_{k+1}}{\sigma_k})$, engineered so that $nW\ge Z$ with equality only when $(b_{ij})$ is a scalar matrix. The heart of the proof is an elliptic differential inequality for $Z$: Proposition 2.3 computes $F^{ij}D_iD_jZ$, and Lemma 4.1 supplies two positivity estimates (one algebraic, based on a quoted inequality from [11], and one from the standard inequality for elementary symmetric means) that make the terms sign-definite. At a maximum point of $W$, where the matrix is scalar, the coefficient of $|Du|^2$ becomes positive, and the strong maximum principle then propagates constancy.

What would settle it

The theorem predicts no nonconstant positive solutions to $\sigma_k(b_{ij})/\sigma_l(b_{ij})=u^{p-1}$ on $S^n$ for $1-(k-l)<p<1$; searching for rotationally symmetric nonconstant solutions reduces this to a one-dimensional ODE, and finding one would refute Theorem 2.1. Independently, the proof's key estimate can be tested directly: for $n=3$, $k=2$, $l=1$ and positive radii, the quoted inequality becomes $2\sigma_2/\sigma_1-3\sigma_3/(2\sigma_2)\le \frac{1}{2}\sigma_1$, and a single failure among positive triples would invalidate Lemma 4.1(i).

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Extended reading notes

Core claim

On the support function $u$ of a strictly convex hypersurface, the self-similar equation becomes $\sigma_k(b_{ij})/\sigma_l(b_{ij}) = u^{p-1}$, where $(b_{ij})=(u_{ij}+u\delta_{ij})$ is the matrix of principal radii, and the desired conclusion is $u\equiv\text{const}$. For $1\le l<k$, the paper proves this by setting $F=(\sigma_k/\sigma_l)^{1/(k-l)}$, so that $F(b_{ij})=u^{p_0}$ with $-1<p_0<0$; $F$ is 1-homogeneous, concave, and has positive-definite derivative. Two auxiliary functions are introduced: $W=u\lambda_{\max}(b_{ij})-\frac{\beta}{2}(u^2+|Du|^2)$ and $Z=uG-\frac{n\beta}{2}(u^2+|Du|^2)$, with $G=\frac{n}{k}(\sigma_1-\frac{(k+1)\sigma_{k+1}}{\sigma_k})$. A maximum-principle argument shows that at a maximum point of $W$ the matrix $(b_{ij})$ is scalar and $Du=0$; then an elliptic inequality for $Z$, obtained from Proposition 2.3 and Lemma 4.1, forces $Z$ to be constant in a neighborhood, hence $W$ is constant and $u$ is constant. In geometric terms, a constant support function corresponds to a sphere. The case $l=0$ is covered by the earlier result in [9].

Load-bearing premise

For the $1\le l<k$ case, the argument depends on an algebraic inequality quoted from [11] (Lemma 2.1 there) bounding $(l+1)\sigma_{l+1}/(l\sigma_l)-(k+1)\sigma_{k+1}/(k\sigma_k)$ by $(1/l-1/k)\sigma_1$; this paper gives no proof of it, and Lemma 4.1(i)—hence the whole maximum-principle step for $Z$—rests on that quoted inequality.

Editorial extensions

If this is right

For any $0\le l<k\le n$ and any $\alpha>1/(k-l)$, every closed strictly convex self-similar shrinker of the quotient flow is a sphere.
In support-function form, the only positive solutions of $\sigma_k(b_{ij})/\sigma_l(b_{ij})=u^{p-1}$ on $S^n$ with $1-(k-l)<p<1$ are constants.
This extends the previously known uniqueness at $\alpha=1/(k-l)$ to the entire open range above it.
The proof requires only strict convexity, so it gives a clean classification of convex self-similar solutions for this family of flows.

Reading between the lines

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

The same $Z$-function construction may prove uniqueness for other 1-homogeneous concave curvature functions once the two inequalities in Lemma 4.1 are checked, since the maximum-principle part of the proof is written for a general $F$.
If the quoted algebraic inequality admits a strengthening, the method might reach exponents below $1/(k-l)$; the current proof is tied to the range where the sign of a particular coefficient becomes positive.
A natural numerical test is to look for rotationally symmetric nonconstant solutions of the support-function equation near the critical exponent $\alpha=1/(k-l)$; the theorem predicts none exist above it, and the boundary case is known by separate arguments.
The result supports the broader expectation that concave, homogeneous curvature flows have only spherical convex self-similar shrinkers, though the mechanism here does not obviously extend to nonconvex or noncompact settings.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The theorem is a real extension: it covers the full range 0≤l1/(k-l), going beyond the α=1/(k-l) cases in Andrews and McCoy, the σ_k^α case in Gao–Li–Ma, and the l=0 case in Chen. The W/Z function strategy is applied carefully, and the maximum-principle argument in Lemma 3.2 is detailed enough to follow. The final coefficient computation at the end of Section 4 checks out under the stated normalization, and Lemma 4.1(ii) is a clean use of Newton's inequality. The paper is well organized and the main ideas are transparent. So credit is due: this is a substantive contribution, not a repackaging.

The soft spots are real but mostly fixable. The biggest one is Lemma 4.1(i): the proof reduces the needed positivity to inequality (15), which is derived solely from Lemma 2.1 of [11], the authors' own unpublished preprint. That lemma is not proved in the paper, and it is load-bearing. If it has hidden hypotheses (e.g., k≥2 or extra convexity), the maximum-principle step for Z collapses. This is exactly the concern you flagged, and it holds up on reading. The authors should either include a proof of the inequality or cite a published source; as written, the paper is not self-contained on a pivotal point.

There are also two smaller issues. First, the derivation of (15) divides by l, so it does not cover l=0; the l=0 case is handled only by an earlier remark citing [9], which is fine but should be stated in the proof of Theorem 2.1. Second, Theorem 2.1 states the range as "1 > p > 1 - k - l," which is a typo: it should be 1 > p > 1 - (k-l), consistent with the −1

As for the stress-test, I think it is accurate. The concern is not manufactured; it is a genuine external dependence. But it is not necessarily fatal. The inequality is plausible and likely true, and a referee could verify it. Still, the paper should not be accepted in its current form without that gap being closed.

Who should read this? Geometric analysts working on curvature flows, convex hypersurfaces, and Michael–Simon-type uniqueness problems. It deserves a serious referee, but the referee should demand a proof of the algebraic inequality and a cleanup of the typos before publication.

Referee Report

2 major / 5 minor

Summary. The paper studies closed, strictly convex self-similar shrinkers in R^{n+1} for curvature flows whose speed is a power of a quotient of elementary symmetric functions of principal curvature, F^α = (σ_k(κ)/σ_l(κ))^α with 0 ≤ l < k ≤ n. Theorem 1.1 asserts that, for α > 1/(k-l), the only such hypersurfaces are round spheres. Passing to the support function, the equation is rewritten as the PDE σ_k(b)/σ_l(b) = u^{p-1} on S^n, where b_{ij} = u_{ij} + u δ_{ij}. The proof combines a W-function maximum principle adapted from Brendle–Choi–Daskalopoulos with a Z-function introduced by the first author, proves at a maximum point of W that the second fundamental form is scalar, and then uses the strong maximum principle to conclude that u is constant. The final coefficient computation at the end of Section 4 is algebraically correct under the stated normalization.

Significance. The theorem fills a genuine gap in the literature: previous uniqueness results covered σ_k^α flows and the 1-homogeneous case α = 1/(k-l), but not general quotient powers. The W/Z method is a nontrivial adaptation to this setting, and the reduction from the geometric equation to the support-function PDE is conceptually clean. If the missing algebraic estimate is supplied, the result is a solid contribution to the uniqueness theory of self-similar solutions to curvature flows. The main weakness is that a key inequality is quoted from a self-cited preprint without proof; until that is fixed, the proof is not fully self-contained at a load-bearing point.

major comments (2)

[Section 4, Lemma 4.1(i)] The proof of inequality (15) is the only support for the nonnegativity of the first two terms in the expression for F^{ij}D_iD_jZ in Proposition 2.3, and it is entirely derived from Lemma 2.1 of the authors' previous preprint [11], which is quoted without proof. This is a load-bearing external dependency: if Lemma 2.1 has hidden hypotheses or is transcribed incorrectly, Lemma 4.1(i) fails and the maximum-principle argument for Theorem 2.1 has no foundation. Please include a self-contained proof of the inequality (or at least of the case needed here) in the paper, or provide a precise reference to a published version with the exact statement.
[Section 4, Lemma 4.1(i), l=0 case] The proof of (15) divides by l when writing the term with i=l+1 as i σ_i/((i-1)σ_{i-1}); for l=0 this term is σ_1/(0·σ_0), so the argument as written excludes l=0. Since Theorem 2.1 is stated for 0 ≤ l < k, this is a gap in the written proof. The authors should state that l=0 is covered by Remark 2.2 and the result of [9], or they should give a separate limiting argument for l=0.

minor comments (5)

[Section 2, transition between equations (9) and (10)] The sentence 'Since For the convenience of discussion, instead of (8), we consider...' is grammatically incomplete; it should be split into two sentences.
[Section 2, Theorem 2.1 reduction] The reduction from Theorem 1.1 to Theorem 2.1 is not fully explicit: for a given equation (7) with indices (k,l), the support-function equation (8) holds for the reversed pair (n-l, n-k) and with p-1 = -1/α, not for the same (k,l). The equivalence is true because both theorems quantify over all pairs, but stating the index reversal and the relation p = 1 - 1/α would prevent confusion.
[Section 3, Lemma 3.2] Lemma 3.2 does not state the value of β in the definition of W; the proof silently uses β = 1+p0. Please make this explicit in the statement.
[Section 3, equation (14)] Inequality (14) is asserted without proof; it follows immediately from F^{ii} > 0, b_{ii} > 0, and b_{11} ≥ b_{ii}, but a one-line justification would improve readability.
[Section 4, Lemma 4.1(ii)] The inequality in part (ii) is attributed to Newton's inequality; a brief explanation of the displayed step would help the reader verify that the constants are correct.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the maximum-principle proof is self-contained, with the only overlapping-author citation being an independent algebraic lemma, not a restatement of the target result.

full rationale

The paper's Theorem 2.1 is proved by a maximum-principle argument on the auxiliary functions W and Z. The decisive estimates in Lemma 4.1 are algebraic consequences of Newton's inequality and of an elementary-symmetric-polynomial inequality cited from Lemma 2.1 of [11]. That cited inequality concerns only σ_k on a positive definite matrix (b_ij), is parameter-free, and does not assume the sphere conclusion or the self-similar equation; it is therefore independent support rather than a circular input. The l=0 boundary case is remanded to [9], also a self-cited external theorem, but it is a separate uniqueness statement for the Lp-Christoffel-Minkowski problem, not an assumption of Theorem 2.1. The derivation from Proposition 2.3 to the strict maximum principle uses only the concavity/convexity of the curvature quotients, the barrier construction, and standard identities, none of which presuppose that u is constant. I found no equation whose conclusion is identical by construction to a fitted parameter or to a self-citation chain; the proof would be vulnerable to a transcription error in [11], but that is a correctness risk, not circularity.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The paper is a proof-based geometry paper and introduces no fitted constants or new physical entities. The only hand-chosen constant is β=1+p0, selected to make the W-function maximum-principle term positive. The proof depends on several external algebraic inequalities from the authors' prior work, most importantly Lemma 2.1 of [11], which is not re-proved here.

free parameters (1)

β = 1 + p0
Chosen by hand in the zeroth-order term of the W and Z functions to make the maximum-principle coefficient at the end of Theorem 2.1 positive (Lemma 4.1 and the final computation in Section 4).

assumptions (4)

domain assumption The algebraic inequality from Lemma 2.1 of [11]: for σ_i the elementary symmetric polynomials, 1/(k(k-1))σ1 - kσ_k/((k-1)σ_{k-1}) + (k+1)σ_{k+1}/(kσ_k) ≥ 0.
Invoked in Lemma 4.1(i) to prove the lower bound on G; it is cited to the authors' own preprint [11] and not proved in this paper.
domain assumption Concavity of F=(σ_k/σ_l)^{1/(k-l)} in (b_ij), and convexity of G=n/k(σ1-(k+1)σ_{k+1}/σ_k).
Used throughout Sections 3 and 4 to get the signs of second-order terms (e.g., -F^{ij,pq}b_{ij1}b_{pq1} ≥ 0). These are standard results from the curvature-flow literature [3,5,11].
standard math The strong maximum principle for second-order elliptic operators with bounded coefficients, applied to LZ=F^{ij}D_iD_jZ - (2/u)F^{ij}u_iD_jZ.
Used at the end of the proof of Theorem 2.1 to conclude that W is constant from local constancy.
standard math Newton's inequalities for elementary symmetric polynomials: σ_{k-1}σ_{k+1} ≤ σ_k^2 (up to combinatorial constants), used to show Σ_i G^{ii} ≤ n in Lemma 4.1(ii).
Classical algebraic fact, cited in Lemma 4.1(ii) without proof.

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Cite this review

Pith. "Pith review of Uniqueness of self-similar solutions to flows by quotient curvatures." pith.science (2026). https://pith.science/paper/P2Y6WCOK

@misc{pith2026190804699,
  author       = {Pith},
  title        = {Pith review of: Uniqueness of self-similar solutions to flows by quotient curvatures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P2Y6WCOK}},
  note         = {Machine review of arXiv:1908.04699}
}

read the original abstract

In this paper, we consider a family of closed hypersurfaces which shrink self-similarly with speed of quotient curvatures. We show that the only such hypersurfaces are shrinking spheres.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

17 extracted references · 16 canonical work pages

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Pith tools

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

[1] [11]

S. Z. Gao, H. Ma, Self-similar solutions of σ α k -curvature ﬂow. arXiv:1611.0758 (2016)

work page arXiv 2016

[2] [9]

Uniqueness of solutions to Lp-Christoffel-Minkowski problem for p<1

L. Chen, Uniqueness of solutions to Lp-Christoﬀel-Minkowski pr oblem. arXiv:1905.11043 (2019)

work page Pith review arXiv 2019

[3] [1]

Pa ciﬁc J

Andrews, B.: Motion of hypersurfaces by Gauss curvature. Pa ciﬁc J. Math. 195 (2000),1-34

work page 2000

[4] [2]

Abresch, J

U. Abresch, J. Langer, The normalized curve shortening ﬂow an d homothetic solutions. J. Diﬀerential Geom. 23, no. 2, 175-196 (1986)

work page 1986

[5] [3]

Andrews, Contraction of convex hypersurfaces in Euclidean space

B. Andrews, Contraction of convex hypersurfaces in Euclidean space. Calc. Var. Partial Diﬀer. Equ. 2, 151171 (1994)

work page 1994

[6] [4]

Andrews, Gauss curvature ﬂow: the fate of the rolling stone s

B. Andrews, Gauss curvature ﬂow: the fate of the rolling stone s. Invent. Math. 138, 151161 (1999)

work page 1999

[7] [5]

Andrews, Pinching estimates and motion of hypersurfaces by curvature functions

B. Andrews, Pinching estimates and motion of hypersurfaces by curvature functions. J. Reine Angew. Math., 608, 17-33 (2007)

work page 2007

[8] [6]

Andrews, P.-F

B. Andrews, P.-F. Guan, L. Ni, Flow by powers of the Gauss curva ture. Adv. Math. 299, 174201 (2016)

work page 2016