Gradient Shrinking Ricci Solitons and Modified Sectional Curvature

Ernani Ribeiro Jr; Hosea Wondo; Xiaodong Cao

REVIEW 3 major objections 4 minor 60 references

Gradient Shrinking Ricci Solitons and Modified Sectional Curvature

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Pith's one-line read Sharp curvature pinching forces complete 4-dimensional shrinking Ricci solitons to be locally Kähler; compact cases reduce to S^4 or CP^2.

desk verdict One clean theorem (Theorem 1) plus a sign error in Theorem 5 that changes the advertised Hitchin–Thorpe constant; Theorems 2–4 are plausible but ride on published black boxes. read the letter →

arxiv 2509.20669 v3 pith:PJCFWERI submitted 2025-09-25 math.DG

Xiaodong Cao , Ernani Ribeiro Jr , Hosea Wondo This is my paper · ORCID

classification math.DG MSC 53C2553C2053E20

keywords gradientRiccisolitonmodifiedsectionalcurvatureself-dualWeyltensorpinchingKähler-RicciHitchin-Thorpeinequalityfour-manifoldrigidity

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 aims to show that four-dimensional gradient shrinking Ricci solitons—self-similar solutions of the Ricci flow—are far more rigid than the unresolved classification problem suggests. Its first theorem proves that when the self-dual part of the Weyl tensor and the scalar curvature obey a sharp two-sided inequality, the soliton must be locally Kähler-Ricci; the product S^2×R^2 attains equality and shows the interval cannot be enlarged. For compact solitons, the paper establishes that a lower bound on a modified sectional curvature, together with a scalar-curvature lower bound, leaves only the round S^4 and the complex projective plane CP^2 as possibilities. It also derives a weighted integral gap for the half-Weyl tensors and a Hitchin–Thorpe type inequality χ > 2.17 τ under weaker integral pinching. If correct, these results convert curvature conditions into topological and complex-geometric conclusions.

What carries the argument

The engine is the drifted Laplacian identity (2.16) for |W+|² on any four-dimensional gradient shrinking Ricci soliton, combined with two sharp algebraic estimates: det W+ ≤ (√6/18)|W+|³ and ⟨(˚Ric⊙˚Ric)+, W+⟩ ≤ (√6/3)|˚Ric|²|W+|. Equality conditions in these estimates detect Kähler structure. For the compact theorems, the paper uses the modified curvature tensor R = R + ½ Hess f ⊙ g and the resulting modified sectional curvature K, whose lower bound, via Proposition 3 from the authors' earlier paper, controls scalar curvature, the drift Laplacian of the potential, and the norms of the half-Weyl tensors. These inputs feed into known gap and rigidity theorems for Einstein four-manifolds and f

What would settle it

For Theorem 1, look for a complete non-Kähler gradient shrinking Ricci soliton with S/(2√6) ≤ |W+| ≤ (1/√6)(2−S/2); the product S^2×R^2 saturates both inequalities, so any strictly interior example is a counterexample. For Theorems 2–5, compute the modified sectional curvature and the quotient χ/τ on the known compact non-Einstein soliton metrics on CP^2#(-CP^2) and CP^2#2(-CP^2); a single example satisfying K≥0.312 and S≥3.694 (or the integral pinching) would falsify the classification and the Hitchin–Thorpe bound.

Watch

Extended reading notes

Core claim

The central discovery is a rigidity mechanism: a Weitzenböck-type formula for |W+|² combines with sharp algebraic eigenvalue bounds so that the pinching S/(2√6) ≤ |W+| ≤ (1/√6)(2−S/2) forces equality in all the estimates, and the equality cases are recognized as Kähler forms (Theorem 1). Building on the same formula and on the modified curvature tensor R+½ Hess f⊙g, the authors derive, for compact solitons with K≥ε and S≥δ, integral estimates that force the Weyl tensor to be harmonic, hence Einstein, and then apply a positive-curvature rigidity theorem to obtain S^4 or CP^2 (Theorems 2–3). A weighted version yields a gap inequality for ∫|W±|²e^{-f}, and a further integration argument gives t

Load-bearing premise

All theorems after Theorem 1 rest on Proposition 3 of the authors' earlier paper [10] — four inequalities converting K≥ε into bounds on scalar curvature, the drift Laplacian, and the half-Weyl norms — and on weighted-Yamabe implications used in Theorems 3 and 5; these are imported without proof, so any hidden normalization, orientation, or sign assumption in them would shift every numerical threshold.

Editorial extensions

If this is right

A four-dimensional complete gradient shrinking Ricci soliton that satisfies the sharp pinching (1.4) must be locally Kähler-Ricci; the boundary cases are exactly S^2×R^2, so the interval is optimal.
Any oriented compact gradient shrinking Ricci soliton with K≥0.312 and S≥3.694 is isometric to S^4 or CP^2, giving a finite classification under pointwise lower curvature bounds.
The same dichotomy holds under the weaker integral pinching ∫|δW+|² ≤ ∫(S/6)|W+|² with K≥0.3069 and S≥3.668, so rigidity persists under averaged conditions.
Under K≥ε and S≥δ (with ε,δ satisfying 21/2<2δ+12ε), the weighted L² norms of the self-dual and anti-self-dual Weyl tensors are bounded by a constant α times ∫S²e^{-f}, a gap estimate in the weighted setting.
If K≥0.186 and the integral pinching holds, any compact oriented four-dimensional gradient shrinking Ricci soliton satisfies χ(M) > (1/0.4613)τ(M) — a Hitchin–Thorpe type inequality stronger than the Einstein one, giving topological obstructions.

Reading between the lines

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

The sharpness of Theorem 1 suggests that the quantity |W+| − S/(2√6) functions as a Kählerity defect; one might track whether this defect contracts under Ricci flow, which would give a dynamical proof of Kähler rigidity for singularities.
The thresholds ε=0.312 and δ=3.694 are computed from algebra plus the black-box inequalities (2.20); if those inequalities are optimal, the thresholds are likely near the exact rigidity boundary, so constructing solitons with ε slightly smaller could test sharpness.
Because the integral pinching used in Theorems 3 and 5 is tied to a weighted Yamabe functional, an independent computation of that functional for known solitons on CP^2♯(-CP^2) or CP^2♯2(-CP^2) would clarify whether the constant 0.186 is essential or an artifact.
The modified sectional curvature mixes the Riemannian curvature with the soliton potential; analogues of these theorems for gradient expanders or steady solitons would require sign changes in the Hess f term, but the same mechanism may apply.

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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

Xiaodong—

Quick take: this is a real but uneven paper. Theorem 1 is a clean, new rigidity statement and its proof is basically sound: the pinching (1.4) is sharp and the maximum principle/cut-off argument works once you insert a missing ρ² factor in the noncompact integration by parts (a typo, not a substantive gap). Theorems 2 and 3 are plausible, and the explicit thresholds are new. But Theorem 5 as printed is not established.

The stress-test note is right. In the proof of (4.39), after reversing orientation, (4.36) should apply to W− with the signature flipped: you get ∫|W−|² > (4/11)π²(2χ−3τ), not (2χ+3τ). Using the printed (2χ+3τ) is exactly what produces the denominator 39. With the correct sign, the constant in (4.39) becomes about 2/3, not 6/13, so the stated χ > 2.1678τ doesn't follow. The theorem can likely be repaired with a weaker constant, but the printed statement is wrong. There is also a minor mismatch: the theorem states ε=0.186 while the proof uses ε=0.184.

I disagree with one point in the reader's report. The concern about Theorem 2 using Catino's theorem as an unconditional inequality is not a real flaw: the inequality ∫|W|² + (5/4)∫|Ric|² ≥ (1/48)∫S² follows from Catino's rigidity by contrapositive. If the strict reverse held, Catino would force a round sphere, and for the round sphere the inequality actually holds. The step is fine, though the paper should have spelled out the case split.

The bigger structural caveat is the reliance on Proposition 3 of [10] for all the constants in Theorems 2–4. That is a published source, so it's acceptable, but the thresholds are only as good as that earlier normalization, and an unstated sign or orientation hypothesis there would shift everything. The paper should at least flag that dependency explicitly.

Bottom line: specialists in Ricci soliton classification will want Theorem 1, and Theorems 2–4 are probably salvageable after some algebra. Theorem 5 needs a corrected constant and a consistent ε. I would send this to a serious referee—it's a competent continuation of a known program, not a crank document—with an explicit request to verify the orientation step and re-derive the constants.

Recommendation: engage with it, but treat Theorem 5's advertised constant as unproven until revised.

Referee Report

3 major / 4 minor

Summary. The paper studies four-dimensional gradient shrinking Ricci solitons. Theorem 1 claims that under the pinching condition S/(2√6) ≤ |W+| ≤ (1/√6)(2 − S/2), a complete soliton is locally a Kähler–Ricci soliton. Theorems 2 and 3 give classification results for compact solitons with a lower bound on the modified sectional curvature K and on the scalar curvature S, concluding that the manifold is isometric to S^4 or CP^2; Theorem 3 also assumes the integral pinching ∫|δW+|² ≤ ∫(S/6)|W+|². Theorem 4 proves a weighted L² estimate for |W±| in terms of S² under K≥ε, S≥δ. Theorem 5 claims a Hitchin–Thorpe type inequality χ(M) > (1/0.4613) τ(M) under K≥0.186 and the same integral pinching. The proofs rely on a Weitzenböck formula from [10], estimates on modified sectional curvature from [10], and rigidity theorems of Catino, Gursky–LeBrun, and Yang.

Significance. If the results are correct, Theorems 1–4 are useful contributions to the classification program for four-dimensional shrinking Ricci solitons. The pinching condition in Theorem 1 is natural in view of Derdziński's identity, and the explicit numerical thresholds in Theorems 2–5 are concrete and checkable. The paper is clearly organized, and the maximum-principle and weighted-integral strategies are appropriate. However, the proof of Theorem 5 contains a sign error in the orientation-reversal step that invalidates the advertised constant, and two other proof steps require repair. As a result, the paper cannot be accepted in its current form.

major comments (3)

[§4.3, Eqs. (4.36)–(4.39)] The orientation-reversal step in the proof of Theorem 5 is invalid. Reversing orientation swaps W+ and W−, so applying (4.36) to the reversed manifold would require the pinching (4.28) for W−, which is not assumed. Moreover, the signature changes sign: the correct lower bound is ∫|W−|² > (4/11)π²(2χ − 3τ), not 2χ + 3τ. Using this in (4.38) gives τ < (18/27 − (88π²/27)(γ/ψ))χ, roughly 0.665χ at the stated parameters, not the 18/39 denominator and 0.4613χ in (4.39). The advertised constant in Theorem 5 is therefore not established. The proof also evaluates constants at ε=0.184 although the theorem states ε=0.186.
[§4.1, before Eq. (4.6)] The proof applies Catino's Theorem 7 as if it gave the unconditional inequality ∫|W|² + (5/4)∫|Ric̊|² ≥ (1/48)∫S². Catino's theorem is a rigidity result: if the reverse inequality holds, the manifold is S^4. The displayed inequality follows only after excluding the round sphere case. The argument needs an explicit case split: if the Catino pinching holds, M is S^4 and the conclusion is already reached; otherwise the strict reverse inequality may be used. As written, the inference leading to (4.6) is logically unjustified.
[§3, after Eq. (3.5)] The step 'substituting equation (3.5) into Proposition 1 yields the equality cases in (2.7) and Lemma 3' is not automatic. If the constant in (3.5) is zero, then Φ = |W+| − S/(2√6) = 0 and inequality (3.1) gives no information, since both sides vanish. Equality in the chain leading to (3.1) is only forced when Φ > 0. The condition |W+| = S/(2√6) alone does not imply the eigenvalue structure required in Lemma 3's equality case. A separate argument or citation is needed before Proposition 2 can be applied.

minor comments (4)

[§2, before (2.9)] Typo: 'Hirzebrush' should be 'Hirzebruch'.
[§2.1, near (2.20)] Typo: 'Propostion' should be 'Proposition'.
[§4.1, Eq. (4.1)] The notation '1/2 ∇∇_f |W+|²' appears to be a typo for '1/2 Δ_f |W+|²'.
[Theorem 5 proof, final paragraph] The proof uses ε=0.184 to compute the final constants, while the theorem states ε=0.186. This discrepancy should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: derivation chain is non-circular, though Theorem 5 contains a non-circular orientation/sign gap.

full rationale

The claimed derivations do not reduce to their own inputs. Theorem 1's pinching (1.4) is an extra hypothesis; the proof uses the Weitzenböck identity (2.16), the algebraic bound (2.7), Lemma 3, and a maximum principle to force equality, then invokes Proposition 2. No displayed identity is the target conclusion by construction. Theorems 2–5 depend on Proposition 3 and weighted-Yamabe criteria quoted from [10] (e.g., 'By [10, Proposition 4.5 and Remark 4.1], assumption (4.10) implies...'). These are self-citations in part, but they are prior published, parameter-free statements whose assumptions (K≥ε, or the W+ pinching) do not include the rigidity/gap/Hitchin-Thorpe conclusions, so per the independence rule they count as real evidence rather than circularity. The constants ε=0.312, δ=3.694, t=0.465 are chosen to satisfy explicit algebraic inequalities (4.6), (4.8), (4.9), (4.40); this is parameter tuning, not fitting a predicted quantity. The one serious defect found is in the proof of Theorem 5: reversing orientation changes the signature to −τ and swaps W+ with W−, so (4.36) cannot be applied to |W−| without the corresponding pinching for W−, and (4.39)'s denominator 39 comes from using 2χ+3τ in the reversed-orientation identity (4.38) instead of 2χ−3τ. This is a mathematical correctness problem in a non-circular derivation, not a circularity.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new entities: the modified sectional curvature is imported from [10]. It does introduce hand-tuned thresholds (epsilon, delta, t) chosen so that algebraic inequalities close, and it relies on several black-box theorems from the same authors' earlier papers and from the literature. The free parameters are constants in proofs, not fitted data.

free parameters (5)

epsilon (Theorem 2, K lower bound) = 1 - sqrt(268/567) ≈ 0.3125
Chosen so that the coefficient comparisons in (4.6) and the identity (1-epsilon)[8 - (1134/67)(1-epsilon)^2] = 0 close exactly, and so that epsilon > (sqrt(1249)-23)/120 ≈ 0.1028 for Yang's rigidity theorem [57].
delta (Theorem 2, S lower bound) = (360/67)sqrt(268/567) ≈ 3.694
Chosen so that the |W|^2 coefficient (5S/2 - 12(1-epsilon) - 4delta/15) is nonnegative exactly at S = delta and the S^2 coefficient becomes -(63/134)(1-epsilon).
t (Theorem 9 interpolation parameter) = 0.465
Chosen to satisfy conditions (4.8) and (4.9) simultaneously with epsilon = 0.3069 and delta = 3.668; at displayed rounding the constraints are borderline, with (4.8) near zero.
epsilon, delta (Theorem 3) = epsilon ≈ 0.3069, delta ≈ 3.668
Rounded solution point in the feasibility region of (4.8) and (4.9); exact values are not given, only a plot for t = 0.465.
epsilon (Theorem 5) = 0.186 stated, 0.184 used in proof's final line
Chosen so that gamma(epsilon) > 0 and psi(epsilon) > 0 and the final numerical bounds are near 1.5008 and 0.4613; the two values disagree between statement and proof.

assumptions (6)

domain assumption Gradient shrinking soliton equation Ric + Hess f = g and the standard identities of Lemma 2, including nabla S = 2Ric(nabla f) and Delta_f S = 2S - 2|Ric|^2.
Defines the object of study and is used throughout Sections 2 to 4.
ad hoc to paper Weitzenböck-type formula Delta_f|W+|^2 = 2|nabla W+|^2 + 4|W+|^2 - 36 det W+ - <(Ric̊ ⊙ Ric̊)+, W+> (Proposition 1).
Central tool from [10] by co-author X. Cao, quoted without proof; drives the proofs of Theorems 1 through 5.
ad hoc to paper Modified sectional curvature bound K >= epsilon implies the four inequalities (2.20): S + 3Delta f >= 12epsilon, S <= 6(1-epsilon), Delta f >= 2(3epsilon - 1), and (1/sqrt(6))(|W+| + |W-|) <= 2(1-epsilon) - S/3 (Proposition 3 of [10]).
Black box from [10]; every constant in Theorems 2 through 5 depends on its exact form.
ad hoc to paper Weighted Yamabe-type propositions of [10]: the pinching conditions (4.10) and (4.28) imply bY_{1,6sqrt(6)}(M) <= 0, and there exists a conformal metric with integral(|W̃+|^2) >= (1/216) integral(S̃^2).
Used without proof in the contradiction arguments of Theorems 3 and 5; if misstated, those two theorems collapse.
ad hoc to paper Catino's gap theorem (Theorem 7) is applied as an unconditional integral inequality in the proof of Theorem 2.
As stated, the inequality fails for S^4; the proof requires an unstated case split that is not written.
domain assumption External rigidity theorems: Yang [57] (Einstein 4-manifolds with K > (sqrt(1249)-23)/120 are S^4 or CP^2), Wu-Wu-Wylie [56] (half-harmonic Weyl implies Einstein), Munteanu-Sesum [41] (L^2_f integrability of |Ric̊|), Chen [18] (positivity of S), Gursky-LeBrun [32] (gap inequality for Einstein manifol
Quoted as the final rigidity inputs in Sections 4.1 and 4.3; the paper does not reprove them.

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Pith. "Pith review of Gradient Shrinking Ricci Solitons and Modified Sectional Curvature." pith.science (2026). https://pith.science/paper/PJCFWERI

@misc{pith2026250920669,
  author       = {Pith},
  title        = {Pith review of: Gradient Shrinking Ricci Solitons and Modified Sectional Curvature},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PJCFWERI}},
  note         = {Machine review of arXiv:2509.20669}
}

read the original abstract

We investigate four-dimensional gradient shrinking Ricci solitons with positive modified sectional curvature. Our first main result shows that if the norm of the self-dual Weyl tensor and the scalar curvature satisfy a certain sharp pinching condition (closely related, in a precise sense, to that of K\"ahler metric), then the soliton is necessarily locally K\"ahler. We further obtain a characterization theorem and a weighted integral gap result for compact gradient shrinking Ricci solitons with bounded modified sectional curvature. In addition, we establish a Hitchin-Thorpe type inequality for compact four-dimensional Ricci solitons, providing new topological constraints on such manifolds.

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Works this paper leans on

60 extracted references · 3 linked inside Pith

[10]

Topol.20 (2016), no

Xiaodong Cao and Hung Tran,The Weyl tensor of gradient Ricci solitons, Geom. Topol.20 (2016), no. 1, 389–436. MR3470717

2016
[1]

Richard Bamler,Structure theory of non-collapsed limits of Ricci flows, ArXiv:2009.03243 (2020)

arXiv 2009
[2]

Bamler, Charles Cifarelli, Ronan J

Richard H. Bamler, Charles Cifarelli, Ronan J. Conlon, and Alix Deruelle,A new com- plete two-dimensional shrinking gradient K¨ ahler-Ricci soliton, Geom. Funct. Anal.34(2024), no. 2, 377–392

2024
[3]

Marcel Berger,Sur quelques vari´ et´ es d’Einstein compactes, Ann. Mat. Pura Appl.53(1961), 89–95. MR0130659

1961
[4]

Besse,Einstein manifolds(2008), xii+516

Arthur L. Besse,Einstein manifolds(2008), xii+516. Reprint of the 1987 edition

2008
[5]

Huai-Dong Cao,Recent progress on Ricci solitons, Recent advances in geometric analysis, Adv. Lect. Math. (ALM), vol. 11, Int. Press, Somerville, MA, 2010, pp. 1–38. MR2648937 FOUR-DIMENSIONAL GRADIENT RICCI SOLITONS 21

2010
[6]

Huai-Dong Cao, Bing-Long Chen, and Xi-Ping Zhu,Recent developments on Hamilton ’s Ricci flow, Surveys in Differential Geometry. Vol. XII. Geometric flows, Surv. Differ. Geom., vol. 12, Int. Press, Somerville, MA, 2008, pp. 47–112. MR2488948

2008
[7]

J.162(2013), no

Huai-Dong Cao and Qiang Chen,On Bach-flat gradient shrinking Ricci solitons, Duke Math. J.162(2013), no. 6, 1149–1169. MR3053567

2013

Show all 60 references

[8]

Reine Angew

Huai-Dong Cao, Ernani Ribeiro Jr, and Detang Zhou,Four-dimensional complete gradient shrinking Ricci solitons, J. Reine Angew. Math.2021(2021), 127-144

2021
[9]

Differ- ential Geom.85(2010), no

Huai-Dong Cao and Detang Zhou,On complete gradient shrinking Ricci solitons, J. Differ- ential Geom.85(2010), no. 2, 175–185. MR2732975

2010
[11]

Xiaodong Cao, Ernani Ribeiro Jr, and Hung Tran,Rigidity of four-dimensional K¨ ahler-Ricci solitons, to appear in Commun. Anal. Geom., ArXiv:2212.05267 [math.DG]

arXiv
[12]

Xiaodong Cao, Biao Wang, and Zhou Zhang,On locally conformally flat gradient shrinking Ricci solitons, Commun. Contemp. Math.13(2011), no. 2, 269–282. MR2794486

2011
[13]

Gursky, and Hung Tran,Curvature of the second kind and a conjecture of Nishikawa, Comment

Xiaodong Cao, Matthew J. Gursky, and Hung Tran,Curvature of the second kind and a conjecture of Nishikawa, Comment. Math. Helv.98(2023), no. 1, 195–216

2023
[14]

Ann.355(2013), no

Giovanni Catino,Complete gradient shrinking Ricci solitons with pinched curvature, Math. Ann.355(2013), no. 2, 629–635. MR3010141

2013
[15]

Math.303(2016), 279–294

,Integral pinched shrinking Ricci solitons, Adv. Math.303(2016), 279–294. MR3552526

2016
[16]

Giovanni Catino, Paolo Mastrolia, and Dario Monticelli,Gradient Ricci solitons with van- ishing conditions on Weyl, J. Math. Pures Appl. (9)108(2017), no. 1, 1–13

2017
[17]

Chang, Matthew J

Sun-Yung A. Chang, Matthew J. Gursky, and Paul C. Yang,A conformally invariant sphere theorem in four dimensions, Advances in Geometric Analysis / Publ. Math. Inst. Hautes ´Etudes Sci.98(2003), 105–143

2003
[18]

Differential Geom.82(2009), no

Bing-Long Chen,Strong uniqueness of the Ricci flow, J. Differential Geom.82(2009), no. 2, 363–382. MR2520796

2009
[19]

Xiuxiong Chen and Yuanqi Wang,On four-dimensional anti-self-dual gradient Ricci solitons, J. Geom. Anal.25(2015), no. 2, 1335–1343. MR3319974

2015
[20]

Reine Angew

Xu Cheng and Detang Zhou,Rigidity of four-dimensional gradient shrinking Ricci soliton, J. Reine Angew. Math.2023(2023), no. 802, 255-274

2023
[21]

Xu Cheng, Ernani Ribeiro Jr, and Detang Zhou,On Euler characteristic and Hitchin-Thorpe inequality for four-dimensional compact Ricci solitons, Proc. Amer. Math. Soc.10(2023), no. 3, 33-45

2023
[22]

Bennet Chow, Peng Lu, and Bo Yang,Lower bounds for the scalar curvatures of noncompact gradient Ricci solitons, C. R. Math. Acad. Sci. Paris349(2011), no. 23-24. MR2861997

2011
[23]

Conlon, Alix Deruelle, and Song Sun,Classification results for expanding and shrinking gradient K¨ ahler-Ricci solitons, Geom

Ronan J. Conlon, Alix Deruelle, and Song Sun,Classification results for expanding and shrinking gradient K¨ ahler-Ricci solitons, Geom. Topol.28(2024), no. 1, 267–351

2024
[24]

3, 405–433

Andrzej Derdzi´ nski,Self-dual K¨ ahler manifolds and Einstein manifolds of dimension four, Compositio Math.49(1983), no. 3, 405–433. MR707181

1983
[25]

,A Myers-type theorem and compact Ricci solitons, Proc. Amer. Math. Soc.134 (2006), no. 12, 3645–3648

2006
[26]

3, 345–367

Manolo Eminenti, Gabriele La Nave, and Carlo Mantegazza,Ricci solitons: the equation point of view, Manuscripta Math.127(2008), no. 3, 345–367. MR2448435

2008
[27]

Topping,On type-I singularities in Ricci flow, Comm

Joerg Enders, Reto M¨ uller, and Peter M. Topping,On type-I singularities in Ricci flow, Comm. Anal. Geom.19(2011), no. 5, 905–922. MR2886712

2011
[28]

Ann.340(2008), no

Manuel Fern´ andez-L´ opez and Eduardo Garc ´ ıa-R ´ ıo,A remark on compact Ricci solitons, Math. Ann.340(2008), no. 4, 893–896

2008
[29]

,On gradient Ricci solitons with constant scalar curvature, Proc. Amer. Math. Soc. 144(2016), 369–378. MR3415603

2016
[30]

Z.269(2011), no

,Rigidity of shrinking Ricci solitons, Math. Z.269(2011), no. 1-2, 461–466. MR2836079

2011
[31]

,Diameter bounds and Hitchin-Thorpe inequalities for compact Ricci solitons, Q. J. Math.61(2010), no. 3, 319–327. MR2672426

2010
[32]

Matthew Gursky and Claude LeBrun,On Einstein manifolds of positive sectional curvature, Ann. Glob. Anal. Geom.17(1999), 315–328. 22 XIAODONG CAO, ERNANI RIBEIRO JR, AND HOSEA WONDO

1999
[33]

Hamilton,The formation of singularities in the Ricci flow, Surveys in differen- tial geometry, Vol

Richard S. Hamilton,The formation of singularities in the Ricci flow, Surveys in differen- tial geometry, Vol. II (Cambridge, MA, 1993), Int. Press, Cambridge, MA, 1995, pp. 7–136. MR1375255

1993
[34]

Differential Geom.9(1974), 435–441

Nigel Hitchin,Compact four-dimensional Einstein manifolds, J. Differential Geom.9(1974), 435–441

1974
[35]

Thomas Ivey,New examples of complete Ricci solitons, Proc. Amer. Math. Soc.122(1994), no. 1, 241–245. MR1207538

1994
[36]

Differential Geom.100(2015), no

Brett Kotschwar and Lu Wang,Rigidity of asymptotically conical shrinking gradient Ricci solitons, J. Differential Geom.100(2015), no. 1, 55–108. MR3326574

2015
[37]

Claude LeBrun,Einstein manifolds, self-dual Weyl curvature, and conformally K¨ ahler ge- ometry, Math. Res. Lett.28(2021), no. 1, 127–144

2021
[38]

Yu Li and Bing Wang,On K¨ ahler Ricci shrinker surfaces, to appear in Acta Mathematica, ArXiv:2301.09784 [math.DG]

arXiv
[39]

London Math

Xue-Mei Li,On extensions of Myers theorem, Bull. London Math. Soc.27(1995), no. 4, 392–396

1995
[40]

Li Ma,Remarks on compact shrinking Ricci solitons of dimension four, C. R. Math. Acad. Sci. Paris351(2013), no. 21-22, 817–823 (English, with English and French summaries). MR3128968

2013
[41]

Ovidiu Munteanu and Natasa Sesum,On gradient Ricci solitons, J. Geom. Anal.23(2013), no. 2, 539–561. MR3023848

2013
[42]

Ovidiu Munteanu and Jiaping Wang,Geometry of shrinking Ricci solitons, Compos. Math. 151(2015), no. 12, 2273–2300. MR3433887

2015
[43]

Differential Geom.106 (2017), no

,Positively curved shrinking Ricci solitons are compact, J. Differential Geom.106 (2017), no. 3, 499–505. MR3680555

2017
[44]

Reine Angew

Aaron Naber,Noncompact shrinking four solitons with nonnegative curvature, J. Reine Angew. Math.645(2010), 125–153. MR2673425

2010
[45]

Lei Ni and Nolan Wallach,On a classification of gradient shrinking solitons, Math. Res. Lett. 15(2008), no. 5, 941–955. MR2443993

2008
[46]

Grisha Perelman,Ricci flow with surgery on three manifolds, ArXiv:math.DG/0303109 (2003)

2003
[47]

Topol.14(2010), no

Peter Petersen and William Wylie,On the classification of gradient Ricci solitons, Geom. Topol.14(2010), no. 4, 2277–2300. MR2740647

2010
[48]

Z.268(2011), 777–790

Stefano Pigola, Michele Rimoldi, and Alberto Setti,Remarks on non-compact gradient Ricci solitons, Math. Z.268(2011), 777–790

2011
[49]

Natasa Sesum,Convergence of the Ricci flow toward a soliton, Comm. Anal. Geom.14 (2006), no. 2, 283–343

2006
[50]

Homare Tadano,An upper diameter bound for compact Ricci solitons with application to the Hitchin-Thorpe inequality. II, J. Math. Phys.59(2018), no. 4, 043507, 3

2018
[51]

Thorpe,Some remarks on the Gauss-Bonnet integral, J

John A. Thorpe,Some remarks on the Gauss-Bonnet integral, J. Math. Mech.18(1969), 779–786

1969
[52]

MR1768112

Gang Tian and Xiaohua Zhu,Uniqueness of K¨ ahler–Ricci solitons, Acta Mathematica184 (2000), 271–305. MR1768112

2000
[53]

,A new holomorphic invariant and uniqueness of K¨ ahler-Ricci solitons, Comment. Math. Helv.77(2002), no. 2, 297–325. MR1915043

2002
[54]

Math.470(2025), Paper No

Hung Tran,K¨ ahler gradient Ricci solitons with large symmetry, Adv. Math.470(2025), Paper No. 110253, 32

2025
[55]

Higher Education PressII(2007), 1-13

Guofang Wei and William Wylie,Comparison geometry for the smooth metric measure spaces, In ICCM. Higher Education PressII(2007), 1-13

2007
[56]

Jia-Yong Wu, Peng Wu, and William Wylie,Gradient shrinking Ricci solitons of half har- monic Weyl curvature, Calc. Var. Partial Differential Equations57(2018), no. 5, Paper No. 141, 15. MR3849152

2018
[57]

142(2000), 435–450

DaGang Yang,Rigidity of Einstein 4-manifolds with positive curvature, Inventiones Math. 142(2000), 435–450

2000
[58]

Fei Yang and Liangdi Zhang,Rigidity of gradient shrinking Ricci solitons, Asian J. Math. 24(2020), no. 4, 533–547

2020
[59]

Zhu-Hong Zhang,Gradient shrinking solitons with vanishing Weyl tensor, Pacific J. Math. 242(2009), no. 1, 189–200. MR2525510 FOUR-DIMENSIONAL GRADIENT RICCI SOLITONS 23

2009
[60]

,A gap theorem of four-dimensional gradient shrinking solitons, Commun. Anal. Geom.28(2020), no. 3, 729–742. (X. Cao)Department of Mathematics, Cornell University, Ithaca, NY 14853 Email address:xiaodongcao@cornell.edu (E. Ribeiro Jr)Departamento de Matem´atica, Universidade F...

2020

Pith tools

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[1] [10]

Topol.20 (2016), no

Xiaodong Cao and Hung Tran,The Weyl tensor of gradient Ricci solitons, Geom. Topol.20 (2016), no. 1, 389–436. MR3470717

2016

[2] [1]

Richard Bamler,Structure theory of non-collapsed limits of Ricci flows, ArXiv:2009.03243 (2020)

arXiv 2009

[3] [2]

Bamler, Charles Cifarelli, Ronan J

Richard H. Bamler, Charles Cifarelli, Ronan J. Conlon, and Alix Deruelle,A new com- plete two-dimensional shrinking gradient K¨ ahler-Ricci soliton, Geom. Funct. Anal.34(2024), no. 2, 377–392

2024

[4] [3]

Marcel Berger,Sur quelques vari´ et´ es d’Einstein compactes, Ann. Mat. Pura Appl.53(1961), 89–95. MR0130659

1961

[5] [4]

Besse,Einstein manifolds(2008), xii+516

Arthur L. Besse,Einstein manifolds(2008), xii+516. Reprint of the 1987 edition

2008

[6] [5]

Huai-Dong Cao,Recent progress on Ricci solitons, Recent advances in geometric analysis, Adv. Lect. Math. (ALM), vol. 11, Int. Press, Somerville, MA, 2010, pp. 1–38. MR2648937 FOUR-DIMENSIONAL GRADIENT RICCI SOLITONS 21

2010

[7] [6]

Huai-Dong Cao, Bing-Long Chen, and Xi-Ping Zhu,Recent developments on Hamilton ’s Ricci flow, Surveys in Differential Geometry. Vol. XII. Geometric flows, Surv. Differ. Geom., vol. 12, Int. Press, Somerville, MA, 2008, pp. 47–112. MR2488948

2008

[8] [7]

J.162(2013), no

Huai-Dong Cao and Qiang Chen,On Bach-flat gradient shrinking Ricci solitons, Duke Math. J.162(2013), no. 6, 1149–1169. MR3053567

2013