On Ricci solitons whose potential is convex

Absos Ali Shaikh; Chandan Kumar Mondal

REVIEW 3 major objections 4 minor 16 references

On Ricci solitons whose potential is convex

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Complete gradient Ricci solitons with convex potentials are Ricci flat and split a line.

desk verdict The concave-potential half is sound; the convex-potential rigidity theorem is not established because Lemma 2.1's integral estimate needs a sign or absolute value that the assumptions don't supply, and the natural repairs make the statement vacuous. read the letter →

arxiv 1908.08303 v3 pith:EA5DLJRD submitted 2019-08-22 math.DG

Chandan Kumar Mondal , Absos Ali Shaikh This is my paper · ORCID

classification math.DG MSC 53C2053C2153C44

keywords RiccisolitongradientconvexpotentialconcaveflatsplittingtheoremscalarcurvatureKillingvectorfield

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 tries to prove a rigidity statement for Ricci solitons whose potential function is convex. It shows that a complete gradient Ricci soliton with non-negative Ricci curvature, a non-constant convex potential with finite weighted Dirichlet energy, and a finite weighted tail integral must be Ricci flat; the potential's gradient is then a Killing field of constant norm, and the manifold splits isometrically as $N\times \mathbb{R}$. It further shows that a non-constant concave potential with bounded Ricci curvature forces the soliton to be non-shrinking, and that with positive Ricci curvature the scalar curvature has at most one critical point. The result matters because it extends classical splitting and rigidity theorems from affine or harmonic functions to a natural class of convex potentials.

What carries the argument

The engine is the interaction of convexity with cutoff functions and integration by parts. A convex function is subharmonic ($\Delta u\ge 0$), and the cutoff functions $\phi_r$ used in the proof satisfy $|\nabla\phi_r|^2\le C/r^2$ and $\Delta\phi_r^2\le C/r^2$; these bounds, together with the finite weighted tail conditions, are meant to make the boundary term $\int u\,\Delta\phi_r^2$ vanish as $r\to\infty$, leaving $\Delta u=0$. The pointwise identity $\tfrac12\Delta|\nabla u|^2=|\nabla^2 u|^2+\mathrm{Ric}(\nabla u,\nabla u)$, valid once $u$ is harmonic, then forces both the Hessian and the Ricci term to vanish under non-negative Ricci curvature. For the concave-potential half, the main tool is the second variation of arc length along a ray, which bounds the integrated Ricci curvature and rules out $\lambda>0$.

What would settle it

Compute the boundary term in the proof of Lemma 2.1 on a complete manifold with non-negative Ricci curvature for a non-constant convex $u$ that satisfies (3) and (4) but is negative somewhere outside every ball. The paper's estimate $\int u\Delta\phi_r^2\le (C/r^2)\int u\to 0$ is only valid when $u\ge 0$ on the annulus; if the direct evaluation of $\lim_{r\to\infty}\int_{M\setminus B(p,r)}u\,\Delta\phi_r^2$ gives a nonzero value while (3) and (4) hold, then $\Delta u\ne 0$, Lemma 2.1 is false, and Theorem 2.5 loses its proof.

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Extended reading notes

Core claim

The central claim is that if $(M,g,u)$ is a complete gradient Ricci soliton with non-negative Ricci curvature, satisfying $\nabla^2 u+\mathrm{Ric}=\lambda g$, and the non-constant convex potential $u$ obeys the weighted Dirichlet condition $\int_{M\setminus B(p,r)}d(x,p)^{-2}|\nabla u|^2<\infty$ and the weighted tail condition $\int_{M\setminus B(p,r)}d(x,p)^{-2}u<\infty$, then $u$ is affine, its Hessian vanishes, $\nabla u$ is a Killing vector field with constant norm, the soliton constant is $\lambda=0$, and the Ricci curvature is zero. The splitting theorem then gives an isometry $M\cong N\times \mathbb{R}$ with $N$ totally geodesic. The paper also claims that a complete gradient soliton with non-constant concave potential and bounded Ricci curvature must satisfy $\lambda\le 0$, and if the Ricci curvature is positive and $\lambda\ge 0$, the scalar curvature has at most one critical point.

Load-bearing premise

The load-bearing premise is that the potential $u$ is non-negative (or at least has controlled absolute value near infinity), because only under that sign condition does the weighted tail integral (4) force the boundary term to vanish and yield $\Delta u=0$, whereas the paper assumes only the weaker signed integral $\int u\,d^{-2}<\infty$.

Editorial extensions

If this is right

With the convex hypotheses, the soliton constant must be $\lambda=0$: the soliton is steady, never shrinking or expanding.
The manifold is isometric to $N\times \mathbb{R}$, so it contains a line and the level sets of $u$ are totally geodesic.
The gradient $\nabla u$ is a Killing vector field of constant norm, so the potential is an affine coordinate on the $\mathbb{R}$ factor.
A harmonic function with finite weighted Dirichlet integral on such a gradient soliton already forces Ricci flatness, as stated in Corollary 2.5.1.
Under a concave potential with bounded Ricci curvature the soliton is non-shrinking; with positive Ricci and $\lambda\ge 0$ it is steady and its scalar curvature has at most one critical point.

Reading between the lines

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

Going beyond the paper, the failure of the sign hypothesis suggests a concrete repair: replace the signed tail condition by $\int_{M\setminus B(p,r)}d(x,p)^{-2}|u|<\infty$, or explicitly assume $u\ge 0$; with that hypothesis the integration-by-parts argument in Lemma 2.1 becomes valid and the rest of Theorem 2.5 follows as written.
The method also suggests a finite-energy splitting principle: non-negative Ricci curvature plus a convex function with finite weighted Dirichlet energy may split off a line whenever the function is affine in a weak sense, connecting the result to broader rigidity theorems for harmonic functions.
Since the proof only uses pointwise upper bounds on Ricci along a ray, the concave-potential theorem might hold under a weaker one-sided bound on Ricci rather than the global boundedness assumed here, though the paper does not pursue that generalization.

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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

Read this before you cite the first theorem. The interesting part is the attempt to force a gradient Ricci soliton with convex potential and nonnegative Ricci to split via weighted Dirichlet and tail conditions. I don't know a prior statement of exactly that combination, so the question is worth thinking about. But the proof of the key lemma (2.1) is broken in a way that matters.

The first misstep is right after (5): since phi^2 = 1 in B(p,r), the text concludes that the integral of Delta u over B(p,r) is zero. That doesn't follow; the annulus contribution is dropped. The later estimate is more serious: 0 <= integral phi^2 Delta u = integral u Delta phi^2 <= (C/r^2) integral u. The left side is nonnegative because Delta u >= 0, but convexity does not make u nonnegative, so the last inequality has no basis. To make it valid you would need integral |u| or u >= 0, and neither is in the assumptions.

If you try the natural repair, you run into a worse problem. On the manifold the theorem predicts, N x R, a nonconstant affine u has |grad u| = c > 0, satisfies (3), but integral d^{-2}|u| diverges logarithmically in the R-factor. So with the absolute-value version of (4), the theorem would be vacuous. Assuming u >= 0 has the same issue. This is not a line-by-line typo; the statement as written is not supported, and the obvious fixes don't give a useful theorem.

The second half is in better shape. Theorem 3.1 uses the standard second-variation inequality along a ray and concavity to rule out lambda > 0. It needs a little limit bookkeeping, but the argument is sound. Theorem 3.3 is an immediate consequence after coupling with the cited lemma on steady solitons. The references are honest and there's no circularity.

Who gets value: specialists in Ricci solitons who want the concave-potential observation. Nobody should rely on the convex-potential splitting theorem until the lemma is repaired or restated. As it stands, I would desk reject; the report can be short. If an editor wants to be generous, a referee could document why Lemma 2.1 fails, but this is not a citable proof.

Referee Report

3 major / 4 minor

Summary. The paper studies complete gradient Ricci solitons and claims two rigidity results. In Section 2, assuming non-negative Ricci curvature and a non-constant convex potential u satisfying the finite weighted Dirichlet integral (3) and the integral condition (4), Lemma 2.1 asserts that the Hessian of u vanishes. The subsequent results then conclude that the manifold is isometric to a product N × R (Theorem 2.4) and that the soliton is Ricci flat with ∇u a Killing vector field of constant norm (Theorem 2.5). A corollary states a harmonic-function version. In Section 3, the authors prove that a non-constant concave potential with bounded Ricci curvature forces the soliton to be non-shrinking (Theorem 3.1), and under 0 < Ric ≤ K the scalar curvature has at most one critical point (Theorem 3.3). The main tools are cutoff functions, the Bochner formula, and external splitting and scalar-curvature theorems.

Significance. If the proofs were correct, Theorem 2.5 would be a clean rigidity statement: complete gradient Ricci solitons with non-negative Ricci curvature and a non-constant convex potential satisfying (3) and (4) would be Ricci flat and split off a line. The statements are concrete and falsifiable, and the proof strategy is natural. The paper does not rely on fitted parameters or self-citations; the main burden is proof rigor rather than circularity. However, the entire classification in Section 2 rests on Lemma 2.1, whose proof contains a sign-error in the integral estimate and a false local Bochner identity. Because these are load-bearing and the most natural repairs either make the statement vacuous or require genuinely new arguments, the announced results are not established as written. Section 3 contains a related but more local limit-passage gap.

major comments (3)

[Lemma 2.1, Eq. (4) and the estimate after Eq. (5)] The proof of harmonicity uses the chain 0 ≤ ∫ φ_r² Δu = ∫ u Δφ_r² ≤ (C/r²) ∫ u → 0. This chain is not justified by the stated hypotheses. Convexity only gives Δu ≥ 0; the function u itself may change sign, and ∫ u Δφ_r² cannot be controlled by (C/r²) ∫ u unless u ≥ 0 or (4) is replaced by an absolute-value condition. Since (4) is stated for u without absolute value, the conclusion Δu = 0 does not follow. The natural repair, replacing u by |u| in (4), is incompatible with the theorem's own conclusion: on the predicted splitting N × R, a non-constant affine potential with nonzero constant gradient satisfies the weighted Dirichlet condition (3), but the |u|-version of (4) diverges logarithmically in the R-factor. Additionally, the line 'Since φ_r ≡ 1 in B(p,r), using (5), we get ∫_{B(p,r)} Δu = 0' is false, because (5) contains an annulus term ∫_{B(2r)\B(p,r)} Δu φ_r² that is not shown to vanish. This gap is load-bearing for Lemma 2.1 and hence for Theorems 2.4 and 2.5.
[Lemma 2.1, Bochner step around Eqs. (6)–(7)] The displayed local identity ∫_{B(p,r)} (|∇²u|² + Ric(∇u,∇u)) = ∫_{B(p,r)} (1/2)|∇u|² Δφ_r² = 0 is incorrect. On a ball, ∫_{B(p,r)} Δ|∇u|² equals a boundary flux, not ∫ |∇u|² Δφ_r²; the latter vanishes only because φ_r ≡ 1 on B(p,r), while the former need not vanish. A global cutoff argument could repair this by bounding the ball integral by the annulus integral, so this particular error is local and fixable, but as written the proof of Eq. (7) is not valid.
[Theorem 3.1, transition from Eq. (11) to Eq. (12)] The proof asserts lim_{t0→∞} ∫_0^{t0} ∇²u(X,X) dt ≤ 0 and then treats lim_{t0→∞} λt0 as +∞. Neither limit is proved to exist. The argument can be repaired: writing A(t0) = ∫_0^{t0} ∇²u(X,X) dt, concavity gives A(t0) ≤ 0, and Eq. (11) gives λt0 ≤ C + A(t0) ≤ C for all t0, contradicting λ > 0. But the manuscript does not present this limsup-based reasoning, and as written the conclusion λ ≤ 0 does not strictly follow.

minor comments (4)

[Abstract and Introduction] The abstract contains grammatical and typographical errors, e.g. 'we have showed' should be 'we have shown', and 'In p articular' has a missing space.
[Throughout] The notation M − B(p,r) should be typeset as M \setminus B(p,r) (or M \smallsetminus B(p,r)) to avoid confusion with the Minkowski difference.
[References] Reference [13] gives the arXiv identifier as 'math.DG/02111159'; the standard identifier for Perelman's entropy paper is math/0211159. Reference [5] gives the year as 1971; the cited article by Fang, Man, and Zhang appeared in 2008.
[Equation (9)] The second-variation inequality (9) is quoted without specifying the variation used; the authors should indicate that one takes φ times parallel unit normal fields along the geodesic, so that the Ricci term emerges after summing.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's conclusions follow from explicit hypotheses plus external splitting, affine-function, and steady-soliton results, with no fitted parameter or self-citation chain forcing the result.

full rationale

The derivation chain in Theorems 2.4 and 2.5 rests on Lemma 2.1, which aims to show that a complete manifold with nonnegative Ricci curvature and a nonconstant convex function satisfying (3) and (4) has vanishing Hessian. The subsequent steps use Lemma 2.2 (affine functions, cited to Sakai) and Theorem 2.3 (splitting theorem, cited to Innami), both external mathematical results. There is no fitted parameter and no quantity is called a prediction that is actually an input. The only self-citation is reference [15] by two of the authors, used together with Yau's paper [16] for the definition of convexity; this is not load-bearing for the main theorem and does not smuggle in an ansatz or forbid alternatives. The proof of Theorem 3.1 uses the second variation of arc length and a standard cutoff estimate, and Theorem 3.3 uses Lemma 3.2 from Guo; again, these are external. Even if the proof of Lemma 2.1 contains an integration-by-parts gap concerning the sign of u, that is a mathematical-rigor concern about the validity of the argument, not a circularity: a faulty or incomplete step is not the same as assuming the conclusion. No displayed equation in the paper reduces the output to the hypothesis by construction, and no prior result of the same authors is invoked as the sole justification of the central claim. Accordingly, the appropriate circularity score is 0.

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

No free parameters or invented entities appear. The proof relies on standard comparison geometry tools and one external lemma. The strong weighted-integral hypotheses are assumptions, not fitted outputs.

assumptions (6)

standard math Smooth convex functions on complete Riemannian manifolds are subharmonic and force noncompactness.
Used in Lemma 2.1 to justify Delta u >= 0 and to conclude M is noncompact. This comes from Greene-Wu and Yau, references [6] and [16].
standard math Cheeger-Colding cutoff functions exist on complete manifolds with the stated bounds |grad phi|^2 <= C/r^2 and |Delta phi| <= C/r^2.
Used throughout Lemma 2.1 and Theorem 2.4 to localize integral identities on balls.
standard math Bochner formula: half of Delta |grad u|^2 equals |Hess u|^2 plus <grad u, grad Delta u> plus Ric(grad u, grad u).
Used after harmonicity is established to estimate the Hessian and Ricci direction.
standard math Innami splitting theorem: a complete manifold with a nonconstant affine function is isometric to N times R.
Used in Theorem 2.4 to pass from vanishing Hessian to product splitting.
standard math Index form inequality: for a minimizing geodesic, integral phi^2 Ric(X,X) <= (n-1) integral |phi'|^2 for test functions vanishing at endpoints.
Used in Theorem 3.1 to obtain a uniform bound on integrated Ricci curvature along a ray.
standard math Guo's theorem: a steady gradient Ricci soliton with positive Ricci curvature has at most one critical point of scalar curvature.
External result [7] used in Theorem 3.3 after proving lambda = 0.

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Pith. "Pith review of On Ricci solitons whose potential is convex." pith.science (2026). https://pith.science/paper/EA5DLJRD

@misc{pith2026190808303,
  author       = {Pith},
  title        = {Pith review of: On Ricci solitons whose potential is convex},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EA5DLJRD}},
  note         = {Machine review of arXiv:1908.08303}
}

read the original abstract

In this paper we consider the Ricci curvature of a Ricci soliton. In particular, we have showed that a complete gradient Ricci soliton with non-negative Ricci curvature possessing a non-constant convex potential function having finite weighted Dirichlet integral satisfying an integral condition is Ricci flat and also it isometrically splits a line. We have also proved that a gradient Ricci soliton with non-constant concave potential function and bounded Ricci curvature is non-shrinking and hence the scalar curvature has at most one critical point.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

16 extracted references · 15 canonical work pages

[1]

Aubin, T., Some nonlinear problems in Riemannian geometry , Springer, 2013

work page 2013
[2]

Cao, H. D. and Zhu, X. P., A complete proof of the Poincare and geometrization conject ures-application of the Hamilton-Perelman theory of the Ricci ﬂow , Asian J. Math., 10 (2006), 165–492

work page 2006
[3]

and Colding, T

Cheeger, J. and Colding, T. H., Lower bounds on Ricci curvature and the almost rigidity of wa rped products, Ann. Math., 144(1) (1996), 189–237

work page 1996
[4]

and Knopf, D., The Ricci ﬂow: an introduction, mathematical surveys and mo nographs, Amer

Chow, B. and Knopf, D., The Ricci ﬂow: an introduction, mathematical surveys and mo nographs, Amer. Math. Soc., 110, 2004

work page 2004
[5]

Q., Man, J

Fang, F. Q., Man, J. W. and Zhang, Z. L., Complete gradient shrinking Ricci solitons have ﬁnite topo logical type. C. R. Acad. Sci. Paris, Ser. I 346(1971), 653–656

work page 1971
[6]

Greene, R. E. and Wu. H., On the subharmonicity and plurisubharmonicity of a geodesi c convex function , Indiana Univ. Math. J. 22(1971), 641–653

work page 1971
[7]

Guo, H., On the Ricci curvature of steady gradient Ricci solitons , J. Math. Anal. Appl., 363 (2010), 497–501. 8 C. K. MONDAL, A. A. SHAIKH

work page 2010
[8]

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Hamilton, R. S., Three-manifolds with positive Ricci curvature , J. Diﬀ. Geom., 17 (1982), 255–306

work page 1982

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[9]

Innami, N., Splitting theorems of Riemannian manifolds , Compositio Math., 47(3) (1982), 237–247

1982
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and Wang, J., Geometry of shrinking Ricci solitons , Compositio Math., 151(12) (2017), 2273–2300

Munteanu, O. and Wang, J., Geometry of shrinking Ricci solitons , Compositio Math., 151(12) (2017), 2273–2300

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[11]

and Wang, J., Positively curved shrinking Ricci solitons , J

Munteanu, O. and Wang, J., Positively curved shrinking Ricci solitons , J. Diﬀ. Geom., 106 (2017), 499–505

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and Wang, J., Conical structure for shrinking Ricci solitons , J

Munteanu, O. and Wang, J., Conical structure for shrinking Ricci solitons , J. Eur. Math. Soc., 19 (2017), 3377–3390

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[13]

Perelman, G., The entropy formula for the Ricci ﬂow and its geometric appli cations, arXiv.org/abs/math.DG/02111159, (2003)

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[14]

Sakai, T., On Riemannian manifolds admitting a function whose gradien t is of constant norm , Koadi. Math. J., 19 (1996), 39–51

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A, Mondal, C

Shaikh, A. A, Mondal, C. K. and Ahmad, I., Non-existence of certain type of convex functions on a Riemannian manifold with a pole , J. Geom. Phys, 140 (2019), 104–110

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[16]

Yau, S.T., Non-existence of continuous convex functions on certain Ri emannian manifolds . Math. Ann. 207 (1974), 269–270. 12Department of Mathematics, University of Burdwan, Golapbag, Burdwan-713104, West Bengal, India E-mail address : 1chan.alge@gmail.com E-mail address : 2a...

1974

Pith tools

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

[1] [1]

Aubin, T., Some nonlinear problems in Riemannian geometry , Springer, 2013

work page 2013

[2] [2]

Cao, H. D. and Zhu, X. P., A complete proof of the Poincare and geometrization conject ures-application of the Hamilton-Perelman theory of the Ricci ﬂow , Asian J. Math., 10 (2006), 165–492

work page 2006

[3] [3]

and Colding, T

Cheeger, J. and Colding, T. H., Lower bounds on Ricci curvature and the almost rigidity of wa rped products, Ann. Math., 144(1) (1996), 189–237

work page 1996

[4] [4]

and Knopf, D., The Ricci ﬂow: an introduction, mathematical surveys and mo nographs, Amer

Chow, B. and Knopf, D., The Ricci ﬂow: an introduction, mathematical surveys and mo nographs, Amer. Math. Soc., 110, 2004

work page 2004

[5] [5]

Q., Man, J

Fang, F. Q., Man, J. W. and Zhang, Z. L., Complete gradient shrinking Ricci solitons have ﬁnite topo logical type. C. R. Acad. Sci. Paris, Ser. I 346(1971), 653–656

work page 1971

[6] [6]

Greene, R. E. and Wu. H., On the subharmonicity and plurisubharmonicity of a geodesi c convex function , Indiana Univ. Math. J. 22(1971), 641–653

work page 1971

[7] [7]

Guo, H., On the Ricci curvature of steady gradient Ricci solitons , J. Math. Anal. Appl., 363 (2010), 497–501. 8 C. K. MONDAL, A. A. SHAIKH

work page 2010

[8] [8]

S., Three-manifolds with positive Ricci curvature , J

Hamilton, R. S., Three-manifolds with positive Ricci curvature , J. Diﬀ. Geom., 17 (1982), 255–306

work page 1982