Pith. sign in

REVIEW 3 major objections 4 minor 1 cited by

Rigidity of Five-dimensional Shrinking Gradient Ricci Solitons with Constant Scalar Curvature

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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

Pith's one-line read Every complete noncompact five-dimensional shrinking gradient Ricci soliton with constant scalar curvature $R=3\lambda$ is a finite quotient of $\mathbb{R}^2\times \mathbb{S}^3$.

desk verdict A likely-true five-dimensional rigidity theorem with a genuinely new Weyl-curvature technique, but the proof needs repair in three places before it is citable. read the letter →

arxiv 2411.10712 v2 pith:QENCYWQ4 submitted 2024-11-16 math.DG

Fengjiang Li , Jianyu Ou , Yuanyuan Qu , Guoqiang Wu This is my paper · ORCID
classification math.DG MSC 53C2153E20
keywords gradientRiccisolitonshrinkingconstantscalarcurvaturerigidityfive-dimensionalweightedLaplacianlevel-setWeyleigenvalueestimate
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 proves that a complete noncompact five-dimensional gradient shrinking Ricci soliton with constant scalar curvature $R=3\lambda$ must be rigid: it is isometric to a finite quotient of $\mathbb{R}^2\times \mathbb{S}^3$. This confirms the five-dimensional $R=3\lambda$ case of the general conjecture that constant scalar curvature should force a shrinker to split as a product of an Einstein manifold and a Gaussian factor. The result matters because shrinking solitons are the self-similar models that appear when Ricci flow develops singularities, so any rigidity statement narrows the list of possible singularity models. After this theorem, only the $R=2\lambda$ case remains open among five-dimensional shrinkers with constant scalar curvature.

What carries the argument

The central object is the weighted Laplacian $\Delta_f=\Delta-\nabla_{\nabla f}$ applied to $u=\lambda_1+\lambda_2$, the sum of the smallest two eigenvalues of the Ricci tensor. The proof's main estimate controls $\Delta_f u$ using three ingredients: the level sets of the isoparametric potential function $f$ (normalised by $|\nabla f|^2=f$), a four-dimensional Gauss-Bonnet-Chern identity that bounds the Weyl curvature of each level set in terms of $|\nabla_{\nabla f}\operatorname{Ric}|^2/f^2$, and a point-picking compactness argument that proves the curvature is bounded. The final step converts the differential inequality into an integral inequality with a carefully chosen weight $h=f+\tfrac32\log f-\tfrac{40}{f}$, forcing $u=0$ outside a compact set and then globally by analyticity.

What would settle it

A concrete way to test the claim is to search for a complete noncompact five-dimensional shrinking gradient Ricci soliton with constant scalar curvature $3\lambda$ whose Ricci tensor is not $(0,0,\tfrac12,\tfrac12,\tfrac12)$; if such a soliton exists, Theorem 1.1 is false. Short of that, examining a blow-up sequence at points where $|\operatorname{Rm}|\to\infty$ and checking whether the rescaled unit balls have volume tending to zero would decide whether the borrowed noncollapsing bound actually holds in dimension five.

Watch

Extended reading notes

Core claim

The central claim, Theorem 1.1, is that the equation $\operatorname{Ric}+\nabla^2f=\lambda g$ on a complete noncompact five-manifold, together with $R=3\lambda$, forces the soliton to be a finite quotient of $\mathbb{R}^2\times \mathbb{S}^3$. The proof shows that $u=\lambda_1+\lambda_2$, the sum of the two smallest Ricci eigenvalues, satisfies a differential inequality whose integral form forces $u=0$ outside a compact set, and analyticity then makes $u=0$ everywhere; the Ricci eigenvalues become $(0,0,\tfrac12,\tfrac12,\tfrac12)$, the radial directions are Ricci flat, and the soliton splits as the required product. Along the way the paper also establishes boundedness of the full Riemannian curvature and the decay $\lambda_1+\lambda_2\to0$ at infinity.

Load-bearing premise

The proof's point-picking step needs the rescaled five-dimensional metrics to have a uniform lower bound on the volume of unit balls; this bound is imported from a paper on Kähler Ricci shrinker surfaces, and if it fails for general five-dimensional shrinkers the blow-up limit could collapse and the splitting argument would not go through.

Editorial extensions

If this is right

  • Among five-dimensional shrinkers with constant scalar curvature, the only remaining unknown case is $R=2\lambda$: $R=0$ and $R=5\lambda$ are Einstein, $R=4\lambda$ is a quotient of $\mathbb{R}\times N^4$, and $R=3\lambda$ is now rigid.
  • Every five-dimensional shrinker covered by the theorem has bounded Riemannian curvature and its Ricci eigenvalues tend to $(0,0,\tfrac12,\tfrac12,\tfrac12)$ at infinity, so the asymptotic geometry is fully pinned down.
  • Because $\nabla\operatorname{Ric}=0$ follows, the de Rham splitting applies and the result rules out any nontrivial topology in the non-quotient case.
  • The proof's mechanism for controlling level-set Weyl curvature through the four-dimensional Gauss-Bonnet-Chern formula is available for other rigidity questions in which the level sets of the potential function are four-dimensional.

Reading between the lines

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

  • A self-contained proof of the uniform noncollapsing bound for five-dimensional shrinkers would remove the proof's main imported ingredient and make the compactness step independent of the Kähler-surface setting.
  • The dimension-specific use of a four-dimensional Gauss-Bonnet identity suggests that the method does not extend verbatim to dimensions $n\ge6$, where level sets have dimension at least five and the Weyl term needs a different control.
  • The same weighted-Laplacian and barrier-integral framework may be the right tool for the remaining $R=2\lambda$ case, provided the level-set Weyl term can be bounded without an a priori curvature bound.
  • A concrete check of whether rescaled unit balls along any curvature blow-up sequence keep a positive volume lower bound would isolate the one step on which the rigidity conclusion depends.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper proves that a five-dimensional complete noncompact gradient shrinking Ricci soliton with constant scalar curvature R=3λ (normalized to λ=1/2, so R=3/2) is isometric to a finite quotient of R^2 × S^3. The proof studies the sum u = λ_1 + λ_2 of the two smallest Ricci eigenvalues. Using barrier-sense computations, the authors derive a differential inequality for u involving the Weyl curvature of the level sets of the potential f; they control the L^2 norm of this Weyl curvature via the four-dimensional Gauss-Bonnet-Chern formula. A point-picking argument is then used to prove boundedness of the Riemannian curvature and decay of u at infinity. Finally, an integral argument forces u = 0 outside a compact set, and analyticity extends this to the whole manifold, yielding rigidity via a splitting argument.

Significance. If correct, the theorem confirms a nontrivial case of Cao's conjecture in dimension five, complementing the four-dimensional rigidity results of Cheng-Zhou and Fernández-López-García-Río. The paper's strengths include a fully self-contained analytic strategy with no fitted parameters, no circular use of the target theorem, and a careful barrier-sense treatment of the eigenvalue sum. The use of the four-dimensional Gauss-Bonnet-Chern formula to convert a Weyl-curvature integral into a decay statement is an interesting new mechanism in this problem. However, the proof as written contains a false elementary inequality in Lemma 5.1, a miscomputed asymptotic bound in Proposition 5.3, and a missing noncollapsing argument in the compactness step of Theorem 5.5. These issues are load-bearing and prevent the paper from being accepted in its current form.

major comments (3)
  1. [Section 5, Lemma 5.1] The proof uses the inequality K_{12} ≤ K_{12}^2 + 1/16 to pass from Proposition 4.2 to the uniform bound |∇Ric|^2 ≤ C(λ_1+λ_2) + C|W^{Σ(s)}|^2 + C. This inequality is false in general: for K_{12} = 1/2, the left side is 1/2 while the right side is 5/16. Since no a priori bound on K_{12} is available at this stage, the step is unjustified. This estimate is used in Proposition 5.3, so the gap is load-bearing for the subsequent curvature bound and decay results.
  2. [Section 5, Proposition 5.3] The proof concludes that ∫_{Σ(s)} |W^{Σ(s)}|^2 dσ ≤ C√s by substituting Vol(Σ(s)) = c√s into the integral ∫_{Σ(s)} C/s dσ. The correct computation gives C/√s, since C/s times the volume c√s equals cC/√s. The stated bound C√s does not tend to zero, and Theorems 5.5 and 5.6 later invoke a decay of the Weyl integral that is only valid with the corrected C/√s. As written, the proposition is false in its stated form, although the intended conclusion would be restored by fixing the arithmetic.
  3. [Section 5, Theorem 5.5] The passage to a smooth limit of the rescaled metrics requires a uniform κ-noncollapsing bound at the scale r_j = |Rm(p_j)|^{-1/2}. The cited [29] is a paper on Kähler Ricci shrinker surfaces and is not applicable to arbitrary five-dimensional shrinkers, and the asserted uniform lower bound on Vol(B(p_i,1)) in the original metric does not imply such a bound at the much smaller rescaled scale. Without this, Hamilton's compactness theorem cannot be invoked, and the splitting M_∞ = R × N^4, the Ricci-flatness of N^4, and the resulting contradiction are not established. The same compactness issue affects the Cheeger-Gromov convergence used in Theorem 5.6. A repair via Perelman's no-local-collapsing theorem is plausible but is not supplied in the paper.
minor comments (4)
  1. [Section 1] There are several typographical errors, including 'Petensen-Wylie' for 'Petersen-Wylie' and 'Moreovre' for 'Moreover'; the title also contains an unintended space in 'CUR V ATURE'.
  2. [Section 5, Theorem 5.5] The sentence 'By the κ noncollapsed theorem in [29]' is misleading because [29] concerns Kähler Ricci shrinker surfaces; if a general noncollapsing statement is intended, it should be stated explicitly and proved or cited precisely.
  3. [Section 6, Proposition 6.1] The cross-reference 'inequality (4.21)' appears to point to the wrong displayed equation; the subsequent argument likely refers to the integral estimate (6.24). This should be corrected.
  4. [Section 2, Theorem 2.2(iv)] The stated mean-curvature formula H(a) = (n−2R−1)√a appears inconsistent with the later computation H = 1/(2√f) in Section 3 for the level sets; the convention for the mean curvature or the statement of the theorem needs to be reconciled.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity. The main rigidity theorem is derived from the shrinker equations and standard external results; the overlapping-author citations supply technical tools or a side remark, not the target conclusion.

full rationale

I walked the derivation chain from Theorem 1.1 back through the curvature estimates, the integral inequality, the point-picking argument, and the cited rigidity tools. No fitted parameter or constructed quantity is renamed as a prediction, and no equation is defined in terms of the conclusion it is supposed to establish. Theorem 1.1 is proved by deriving a differential inequality for lambda_1 + lambda_2 (Proposition 3.7), a pointwise bound on |Ric|^2 (Proposition 4.2), a decay estimate for the level-set Weyl curvature (Proposition 5.3), boundedness of the Riemannian curvature (Theorem 5.5), decay of lambda_1 + lambda_2 (Theorem 5.6), and finally the integral inequality forcing lambda_1 + lambda_2 = 0 (Propositions 6.1 and 6.2). Each of these steps is computed from the shrinker equation and previously established external results such as [23], [37], and [19], rather than from the assumption that the manifold is a quotient of R^2 x S^3. The self-citations in the paper are not load-bearing in a circular way: [40] supplies the method of applying Delta_f to the sum of the smallest Ricci eigenvalues in the four-dimensional case, [43] supplies a technical barrier-to-distribution integration lemma, and [27] is mentioned only in a remark about the R = 2 lambda case, which is not the main theorem. The most suspicious step, the compactness argument in Theorem 5.5, invokes the kappa-noncollapsing theorem from [29]. Even if the scale of the cited noncollapsing statement is mismatched with the rescaled radius |Rm(p_j)|^{-1/2}, that is a mathematical gap or an unproved hypothesis about local volume growth, not a circular reduction: the quoted result is not the target theorem and is not being assumed in disguise. Thus, under the standing rules, the paper has no demonstrated circularity, only minor self-citation and a possible rigor gap, so the circularity score is 1.

Assumptions & free parameters 0 free parameters · 10 assumptions · 0 invented entities

The proof uses no fitted parameters and postulates no new objects. It leans on a chain of external classification results and standard geometric analysis tools; the main self-citations ([40], [43]) supply technical lemmas rather than the target conclusion.

assumptions (10)
  • standard math Ricci soliton identities (2.3)-(2.7), including Δ_f R = R - 2|Ric|² and the commutation formulas
    Basic identities of gradient Ricci solitons; invoked throughout Sections 2-4.
  • domain assumption Cao-Zhou potential estimate (Theorem 2.1): f is comparable to squared distance
    Gives properness and growth of f, used for integration by parts at infinity.
  • domain assumption Cheng-Zhou isoparametric structure (Theorem 2.2): level sets, volume formulas, dim(M_-)=2R
    Provides the volume growth of level sets used in the Weyl bound.
  • domain assumption Barrier-sense to distribution equivalence (Theorem 2.3 and Proposition 2.5 from [43])
    Legitimizes integration by parts for the eigenvalue-sum inequality; [43] shares an author with the present paper.
  • domain assumption Nonnegative Ricci curvature for R=(n-2)λ (Fernández-López and García-Río [23])
    Gives λ_1=0 and the eigenvalue ordering; a critical external input.
  • standard math Gauss-Bonnet-Chern formula in dimension 4
    Relates the Weyl curvature integral to Euler characteristic and Ricci terms in Lemma 5.2.
  • domain assumption κ-noncollapsing and Hamilton/Cheeger-Gromov compactness for the rescaled sequence
    Needed in Theorem 5.5 to extract a smooth limit M_∞ = R×N⁴; citation to [29] is questionable.
  • domain assumption Classification of complete locally conformally flat manifolds with nonnegative Ricci curvature (Zhu [46], Carron-Herzlich [13])
    Ruled out limits in Theorem 5.6; assumes the classification applies to the limit N⁴ which may have π_1=Z.
  • domain assumption Yamabe equation classification on R⁴ (Caffarelli-Gidas-Spruck [5], Chen-Li [16])
    Shows no complete metric with scalar curvature 3/2 conformal to Euclidean space.
  • domain assumption Real analyticity of complete gradient shrinking Ricci solitons
    Used to extend λ_1+λ_2=0 and ∇Ric=0 from an exterior set to all of M.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Rigidity of Five-dimensional Shrinking Gradient Ricci Solitons with Constant Scalar Curvature." pith.science (2026). https://pith.science/paper/QENCYWQ4

@misc{pith2026241110712,
  author       = {Pith},
  title        = {Pith review of: Rigidity of Five-dimensional Shrinking Gradient Ricci Solitons with Constant Scalar Curvature},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QENCYWQ4}},
  note         = {Machine review of arXiv:2411.10712}
}
abstract

Let $(M, g, f)$ be a $5$-dimensional complete noncompact gradient shrinking Ricci soliton with the equation $Ric+\nabla^2f= \lambda g$, where $\text{Ric}$ is the Ricci tensor and $\nabla^2f$ is the Hessian of the potential function $f$. We prove that it is a finite quotient of $\mathbb{R}^2\times \mathbb{S}^3$ if $M$ has constant scalar curvature $R=3 \lambda$.

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. Rigidity of Five-Dimensional shrinking gradient Ricci solitons

    math.DG 2025-06 conditional novelty 7.0 of 10

    A five-dimensional shrinking gradient Ricci soliton with constant scalar curvature 1 and bounded curvature must be a finite quotient of R^3 × S^2.

Reference graph

Works this paper leans on

46 extracted references · 19 canonical work pages · cited by 1 Pith paper

  1. [29]

    Y. Li, B. Wang, On K¨ ahler Ricci shrinker surfaces, arXiv:2301.09784

    arXiv

  2. [1]

    R. H. Bamler, C. Cifarelli, R. J. Conlon, A. Deruelle, A new complete two-dimensional shrinking gradient K¨ ahler-Ricci soliton, Geom. Funct. Anal., 34(2024), no.2, 377-392

    work page 2024

  3. [2]

    B. L. Chen, Strong uniqueness of the Ricci flow , J. Differential Geom. 82 (2009), no. 2, 363-382

    work page 2009

  4. [3]

    Brendle, Rotational symmetry of self-similar solutions to the Ricci flow , Invent

    S. Brendle, Rotational symmetry of self-similar solutions to the Ricci flow , Invent. Math. 194 (2013), no. 3, 731-764

    2013

  5. [4]

    Brendle, Rotational symmetry of Ricci solitons in higher dimensions , J

    S. Brendle, Rotational symmetry of Ricci solitons in higher dimensions , J. Differential Geom. 97 (2014), no. 2, 191-214. 40 FENGJIANG LI, JIANYU OU, YUANYUAN QU, AND GUOQIANG WU

    work page 2014

  6. [5]

    Caffarelli, B

    L. Caffarelli, B. Gidas, J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth , Comm. Pure Appl. Math. 42 (1989), no. 3, 271-297

    work page 1989

  7. [6]

    H. D. Cao, Existence of gradient K¨ ahler-Ricci solitons, Elliptic and parabolic methods in geometry (Minneapolis, MN, 1994), 1-16

    1994

  8. [7]

    H. D. Cao, Q. Chen, On locally conformally flat gradient steady Ricci solitons , Trans. Amer. Math. Soc. 364 (2012), no. 5, 2377-2391

    2012

Show all 46 references
  1. [8]

    H. D. Cao, Q. Chen, On Bach-flat gradient shrinking Ricci solitons , Duke Math. J. 162 (2013), no. 6, 1149-1169

    2013

  2. [9]

    H. D. Cao, B. L. Chen, X. P. Zhu, Recent developments on Hamilton ’s Ricci flow, Surveys in differential geometry. Vol. XII. Geometric flows, 47-112

  3. [10]

    H. D. Cao, D. T. Zhou, On complete gradient shrinking Ricci solitons , J. Differential Geom. 85 (2010), no. 2, 175-185

    2010

  4. [11]

    H. D. Cao, J. M. Xie, Four-dimensional complete gradient shrinking Ricci solitons with half positive isotropic curvature , Math. Z., 305 (2023), no.2, 22 pp

    2023

  5. [12]

    X. D. Cao, B. Wang, Z. Zhang, On locally conformally flat gradient shrinking Ricci solitons, Commun. Contemp. Math. 13 (2011), no. 2, 269-282

    2011

  6. [13]

    Carron, M

    G. Carron, M. Herzlich, Conformally flat manifolds with nonnegative Ricci curvature Compos. Math. 142 (2006), no. 3, 798-810

    2006

  7. [14]

    R. J. Conlon, A. Deruelle, S. Sun, Classification results for expanding and shrinking gradient K¨ ahler-Ricci solitons, Geem. Topol. 28(2024), no. 1, 267- 351

    2024

  8. [15]

    Cheeger, D

    J. Cheeger, D. Gromoll, The splitting theorem for manifolds of nonnegative Ricci curvature, J. Differential Geometry 6 (1971/72), 119-128

    1971

  9. [16]

    W. X. Chen, C. M. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J. 63 (1991), no. 3, 615-622

    1991

  10. [17]

    X. X. Chen, Y. Q. Wang, On four-dimensional anti-self-dual gradient Ricci solitons, J. Geom. Anal. 25 (2015), no. 2, 1335-1343

    2015

  11. [18]

    B. L. Chen, X. P. Zhu, Uniqueness of the Ricci flow on complete noncompact manifolds, J. Differential Geom.,74(2006), no. 1, 119-154

    2006

  12. [19]

    Cheng, D

    X. Cheng, D. T. Zhou, Rigidity of Four-dimensional gradient shrinking Ricci solitons, J. Reine Angew. Math., 802 (2023), no. 2, 255-274

    2023

  13. [20]

    Cifarelli, R

    C. Cifarelli, R. J. Conlon, A. Deruelle, On finite time Type-I singularities of the K¨ ahler-Ricci flow on compact K¨ ahler surfaces, arXiv:2203.04380v3

    arXiv

  14. [21]

    Feldman, T

    M. Feldman, T. Ilmanen, D. Knopf, Rotationally symmetric shrinking and expanding gradient K¨ ahler-Ricci solitons, J. Differential Geom. 65 (2003), no. 2, 169-209

    2003

  15. [22]

    P. Guan, P. Lu, Y. Xu, A rigidity theorem for codimension one shrinking gradient Ricci solitons in Rn+1, Calc. Var. Partial Differential Equations, 54 (2015), no. 4, 4019-4036

    2015

  16. [23]

    Fern´ andez-L´ opez, E

    M. Fern´ andez-L´ opez, E. Garc ´ ıa-R ´ ıo,On gradient Ricci solitons with constant scalar curvature, Proc. Amer. Math. Soc. 144 (2016), no. 1,369-378

    2016

  17. [24]

    R. S. Hamilton, Three-manifolds with positive Ricci curvature , J. Differential Geom. 17 (1982), no. 2, 255-306

    1982

  18. [25]

    Koiso, On rotationally symmetric Hamilton ’s equation for K¨ ahler-Einstein metrics, Recent topics in differential and analytic geometry, 327-337

    N. Koiso, On rotationally symmetric Hamilton ’s equation for K¨ ahler-Einstein metrics, Recent topics in differential and analytic geometry, 327-337. 5-DIM RICCI SHRINKER WITH CSC 41

  19. [26]

    Kotschwar, On rotationally invariant shrinking Ricci solitons , Pacific J

    B. Kotschwar, On rotationally invariant shrinking Ricci solitons , Pacific J. Math. 236 (2008), no. 1, 73-88

    2008

  20. [27]

    F. J. Li, J. Y. Ou, Y. Y. Qu, G. Q. Wu, Rigidity of Five-Dimensional shrinking gradient Ricci solitons, arXiv:2506.00887

    arXiv

  21. [28]

    Y. Li, B. Wang, Heat kernel on Ricci shrinkers , Calc. Var. Partial Differential Equations 59 (2020), no. 6, Paper No. 194, 84 pp

    2020

  22. [30]

    Eminenti, G

    M. Eminenti, G. LaNave, C. Mantegazza, Ricci solitons: the equation point of view, Manuscripta Math. 127 (2008), no. 3, 345-367

    2008

  23. [31]

    Munteanu, J

    O. Munteanu, J. P. Wang, Geometry of shrinking Ricci solitons , Compos. Math. 151 (2015), no. 12, 2273-2300

    2015

  24. [32]

    Munteanu, J

    O. Munteanu, J. P. Wang, Positively curved shrinking Ricci solitons are com- pact, J. Differential Geom. 106 (2017), no. 3, 499-505

    2017

  25. [33]

    Naber, Noncompact shrinking four solitons with nonnegative curvature , J

    A. Naber, Noncompact shrinking four solitons with nonnegative curvature , J. Reine Angew. Math. 645 (2010), 125-153

    2010

  26. [34]

    L. Ni, N. Wallach, On a classification of gradient shrinking solitons , Math. Res. Lett. 15 (2008), no. 5, 941-955

    2008

  27. [35]

    Perelman, The entropy formula for the Ricci flow and its geometric appli- cations, arXiv:math/0211159v1

    G. Perelman, The entropy formula for the Ricci flow and its geometric appli- cations, arXiv:math/0211159v1

    arXiv

  28. [36]

    Perelman, Ricci flow with surgery on three-manifolds , arXiv: math/0303109

    G. Perelman, Ricci flow with surgery on three-manifolds , arXiv: math/0303109

    arXiv

  29. [37]

    Petersen, W

    P. Petersen, W. Wylie, On the classification of gradient Ricci solitons , Geom. Topol. 14 (2010), no. 4, 2277-2300

    2010

  30. [38]

    Petersen, W

    P. Petersen, W. Wylie, Rigidity of gradient Ricci solitons , Pacific J. Math. 241 (2009), no. 2, 329-345

    2009

  31. [39]

    Pigola, M

    S. Pigola, M. Rimoldi, A. Setti, Remarks on non-compact gradient Ricci soli- tons, Math. Z. 268 (2011), no. 3-4, 777-790

    2011

  32. [40]

    J. Y. Ou, Y. Y. Qu, G. Q. Wu, Some rigidity results on shrinking gradient Ricci soliton, arXiv:2411.06395

    arXiv

  33. [41]

    X. J. Wang, X. H. Zhu, K¨ ahler-Ricci solitons on toric manifolds with positive first Chern class , Adv. Math. 188 (2004), no. 1, 87-103

    2004

  34. [42]

    G. Q. Wu, S. J. Zhang, Remarks on shrinking gradient K¨ ahler-Ricci solitons with positive bisectional curvature , C. R. Math. Acad. Sci. Paris 354 (2016), no. 7, 713-716

    2016

  35. [43]

    G. Q. Wu, J. Y. Wu, Four dimensional shrinkers with nonnegative Ricci cur- vature, arXiv:2505.02315

    arXiv

  36. [44]

    P. Wu, J. Y. Wu, W. Wylie, Gradient shrinking Ricci solitons of half harmonic Weyl curvature, Calc. Var. Partial Differential Equations 57 (2018), no. 5, Art. 141, 15 pp

    2018

  37. [45]

    Z. H. Zhang, Gradient shrinking solitons with vanishing Weyl tensor , Pacific J. Math. 242 (2009), no. 1, 189-200

    2009

  38. [46]

    S. H. Zhu, The classification of complete locally conformally flat manifolds of nonnegative Ricci curvature, Pacific J. Math. 163 (1994), no. 1, 189-199. 42 FENGJIANG LI, JIANYU OU, YUANYUAN QU, AND GUOQIANG WU (Fengjiang Li) Mathematical Science Research Center, Chongqing Uni...

    1994

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.