A note on a diffeomorphism criterion via long-time Ricci flow

Shaochuang Huang; Zhuo Peng

REVIEW 3 major objections 3 minor 1 cited by

A note on a diffeomorphism criterion via long-time Ricci flow

T0 review · 3 major / 3 minor · reviewed 2026-08-05 · deepseek-v4-flash

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

Pith's one-line read A long-time Ricci flow with 1/t Ricci decay and √t injectivity growth is diffeomorphic to R^n.

desk verdict Genuinely new diffeomorphism criterion with a clean dimension-4 application, but Theorem 1.1 as printed states the wrong injectivity-radius hypothesis (β/√t instead of β√t), so the theorem overclaims until that typo is fixed. read the letter →

arxiv 2509.05802 v1 pith:FF6LJZU6 submitted 2025-09-06 math.DG

Shaochuang Huang , Zhuo Peng This is my paper · ORCID

classification math.DG MSC 53E20

keywords Ricciflowlong-timesolutiondiffeomorphismcriterioninjectivityradiussmallcurvatureconcentrationnonnegativemaximalvolumegrowthspace-timeexponentialmap

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 note proves a topological rigidity criterion: a complete manifold that admits a long-time Ricci flow whose Ricci curvature decays like -ψ/t (ψ<1/2) and whose injectivity radius grows like √t must be diffeomorphic to R^n. The theorem statement writes the injectivity condition as β/√t, but the proofs and Remark 1.1 show the operative assumption is the square-root growth (or βt^κ with κ>ψ). Two self-contained proofs are given, one by matching exponential-coordinate charts through time and one through a space-time exponential map on the level set of the normalized distance. The criterion upgrades known small-curvature-concentration results from homeomorphism to diffeomorphism in dimension 4, gives a Ricci-flow proof that 3-dimensional nonnegative-Ricci manifolds with maximal volume growth are R^3, and yields a Sobolev-constant version in all dimensions n≥3.

What carries the argument

The space-time exponential map Exp(v,t) = (exp_{x0,g(t)} v, t), together with the time-dependent distance distortion estimate d_{g(t')}(x,x0) ≤ d_{g(t)}(x,x0)(t'/t)^ψ. Because ψ<1/2, the normalized distance d/√t decreases in t along each fixed point, so the hypersurface M' = {d_{g(t)}(x,x0) = (β/2)√t} is a graph over M; injectivity-radius control makes Exp a diffeomorphism near M', and radial projection identifies M' with R^n. Method 1 instead uses the vector field ∂_t (exp_{x0,g(t)}^{-1}(x)) and a cutoff flow to build explicit diffeomorphisms between time slices.

What would settle it

Perform the containment check in (2.1) with inj(g(t_i)) = β/√t_i: Method 1's first inclusion requires B_{t_i}(R_i) to lie inside the domain where the exponential map is a diffeomorphism, but R_i ≈ (β/2)√t_i is a factor of t_i larger than the guaranteed injectivity radius for large i, so the inequality fails. That calculation settles whether the proof as written establishes the stated theorem.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 1.1: if (M^n,g(t)) is a smooth complete long-time Ricci flow with Ric(g(t)) ≥ -ψ/t for 0<ψ<1/2 and inj(g(t)) ≥ β/√t for β>0, then M^n is diffeomorphic to R^n. Both proofs, however, invoke the injectivity-radius condition at the scale of √t: Method 1 needs the exponential map to be a diffeomorphism on balls of radius comparable to β√t, and Method 2 needs it on {|v| ≤ (3β/4)√t}; Remark 1.1 states the intended general form inj(g(t)) ≥ βt^κ with ψ<κ. The key mechanism is that the normalized distance d_{g(t)}(x,x0)/√t is monotone in time: the Ricci lower bound yields d_{g(t')}(x,x0) ≤ d_{g(t)}(x,x0)(t'/t)^ψ, so with ψ<1/2 each point x≠x0 crosses the hypersurfa

Load-bearing premise

The proof needs the space-time exponential map to be a diffeomorphism on balls of radius comparable to √t; the injectivity-radius lower bound β/√t stated in the theorem does not provide that, and if the true injectivity radius only decays like t^{-1/2}, the construction collapses.

Editorial extensions

If this is right

Corollary 1.1: for n≥4, a complete non-compact manifold with bounded volume growth, a lower bound on local entropy, Ricci bounded below, and sufficiently small L^{n/2} curvature concentration is diffeomorphic to R^n, upgrading the homeomorphism conclusion of [3] and [17] to diffeomorphism in dimension 4.
Corollary 1.2: every 3-dimensional complete non-compact manifold with nonnegative Ricci curvature and maximal volume growth is diffeomorphic to R^3, proved here by Ricci flow rather than minimal-surface classification.
Corollary 1.3: if a complete bounded-curvature manifold satisfies a W^{1,2}-Sobolev inequality and has L^{n/2} curvature small relative to the inverse Sobolev constant, then it is diffeomorphic to R^n.
Remark 1.1: the same proof works for injectivity radius lower bounds of the form βt^κ with κ>0, provided the curvature decay exponent ψ is less than κ.
Remark 1.2: the 3-dimensional nonnegative-Ricci conclusion is sharp in dimension: the Eguchi-Hanson metric shows the analogous statement fails in higher dimensions without additional assumptions.

Reading between the lines

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

The scaling mismatch between the printed β/√t and the β√t used in both proofs means the theorem as literally stated is not established by the arguments given: if the injectivity radius truly decays like t^{-1/2}, the space-time exponential map on the required region is not guaranteed to be a diffeomorphism.
The same two methods should work for any time-dependent radius function a(t) with a(t)/√t strictly monotone and a(t) below the injectivity radius at each time, so the criterion is likely a special case of a more general monotonicity principle.
The κ>0 variant indicated in Remark 1.1 suggests the criterion extends to flows with Ricci ≥ -ψ/t and inj ≥ βt^κ for any κ>ψ, which would cover algebraic curvature decay rates other than exactly 1/t.
Corollary 1.2 uses only nonnegative Ricci and maximal volume growth in dimension 3; the Eguchi-Hanson example noted by the authors shows the dimensional ceiling is real, so the criterion is not vacuous.

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.

Desk Editor's Note

read the letter

First, the useful core: the paper gives a sufficient condition for a complete long-time Ricci flow to force the underlying manifold to be diffeomorphic to R^n. Theorem 1.1 is not in the literature in this form, and the proof is a transparent adaptation of He–Lee and Wang. The main payoff is Corollary 1.1, which upgrades the known homeomorphism conclusion in dimension 4 to diffeomorphism; that is a real, if subfield-level, advance. The exposition is honest about Remark 1.2 being known to experts, and the citation pattern is clean — the self-citation to [3] is to the result being improved, not a load-bearing dependency. No circularity.

Now the soft spot, and it is load-bearing: the stated hypothesis (ii), inj(g(t)) ≥ β/√t, is not what the proofs use. Method 1 needs the exponential map to be a diffeomorphism on balls of radius comparable to β√t_i; Method 2 explicitly asserts Exp|_Ω is a diffeomorphism for Ω = {d ≤ (3β/4)√t}, which again requires an injectivity radius of order √t. The printed β/√t is far smaller than these radii for large t. Remark 1.1 compounds the issue by saying the condition can be β t^κ with κ > 0 — which is also incompatible with β/√t. So the intended theorem is with inj ≥ β√t (equivalently β t^κ, κ > 0, and ψ < κ). As written, Theorem 1.1 is not established.

That said, this looks like a typo rather than a deep flaw. Every application in Section 3 produces inj ≥ β√t, and the proofs go through verbatim once condition (ii) is corrected. The stress-test concern is accurate; I do not see a way to make the stated β/√t condition suffice. A referee should catch this immediately, and the fix is a one-line change to the theorem statement.

Who is this for? Geometric analysis readers interested in Ricci-flow proofs of topological rigidity. It is a solid extension of known techniques, not a paradigm shift. Corollary 1.2 is essentially a known fact (Liu's classification), so the novelty there is thin, but Corollary 1.1 carries the paper.

Verdict: deserve a serious referee. The central idea is sound, the applications are meaningful, and the flaw is fixable. I would engage with it, but I would not cite Theorem 1.1 in its current form — only after the hypothesis is corrected.

Recommendation: send to peer review, with the clear instruction that the injectivity-radius condition in Theorem 1.1 must be corrected to β√t (or β t^κ, κ>0, with ψ<κ) before acceptance.

Referee Report

3 major / 3 minor

Summary. The paper proposes a diffeomorphism-to-R^n criterion for complete non-compact manifolds that admit a long-time Ricci flow. Theorem 1.1 states that if (M^n, g(t)) is a smooth complete solution on [0,∞) with Ric(g(t)) ≥ −ψ/t for some 0<ψ<1/2 and inj(g(t)) ≥ β/√t for some β>0, then M is diffeomorphic to R^n. Two proofs are given: Method 1 adapts He–Lee's isotopy argument via time-dependent exponential maps, and Method 2 adapts Wang's space-time hypersurface construction. The paper then derives three applications: Corollary 1.1 upgrades the homeomorphism conclusion of Chan–Huang–Lee and Martens to diffeomorphism in dimension 4; Corollary 1.2 gives a Ricci-flow proof of the fact that a 3-dimensional complete non-compact manifold with nonnegative Ricci curvature and maximal volume growth is diffeomorphic to R^3; Corollary 1.3 gives a Sobolev-constant version of the small-curvature-concentration criterion. The central claim is therefore a useful black-box criterion, but the version stated in Theorem 1.1 is not what the proofs establish.

Significance. If Theorem 1.1 is corrected to the injectivity-radius bound actually used, the paper provides a clean and potentially widely applicable diffeomorphism criterion for long-time Ricci flows. The applications are nontrivial and would be valuable: Corollary 1.1 answers the dimension-4 question left open in [3] and [17], and Corollary 1.3 sharpens a known integral-curvature rigidity result. The proofs are assembled from standard tools (distance distortion under Ricci flow, exponential-map embeddings, isotopy by cutoff vector fields, space-time hypersurfaces) and the applications rely on previously established existence and injectivity-radius estimates in [3], [17], [22], [4], [7]. The argument is not circular: Theorem 1.1 is proved from standard geometric-analysis ingredients, and the applications independently supply the needed flow with the strong injectivity bound. The main deficiency is a statement–proof mismatch in the injectivity-radius hypothesis, which affects the central theorem as printed.

major comments (3)

[Theorem 1.1, condition (ii); Section 2, Methods 1 and 2] The stated hypothesis inj(g(t)) ≥ β/√t is weaker than what both proofs require. In Method 2, the set Ω is defined as {d_{g(t)}(x,x0) ≤ (3β/4)√t} and the text asserts 'by the injectivity condition (ii) again, Exp|_Ω is a diffeomorphism.' This requires exp_{x0,g(t)} to be injective on the ball of radius (3β/4)√t, i.e. inj(g(t)) ≥ (3β/4)√t. For t > 4/3 the stated bound β/√t is strictly smaller than (3β/4)√t, so the assertion is unjustified. In Method 1, R_i = (1/2 − 1/(2i+1)) β√t_i is used as a radius on which exp_t is a diffeomorphism, which again requires inj(g(t_i)) ≳ β√t_i/2, far above β/√t_i for large i. The proof therefore establishes the theorem only under the stronger condition inj(g(t)) ≥ β√t (or a comparable positive-power lower bound). The theorem statement must be corrected accordingly.
[Remark 1.1] Remark 1.1 says that if condition (ii) is changed to inj(g(t)) ≥ βt^κ for some κ > 0, then the curvature condition only needs 0 < ψ < κ. This is internally inconsistent with the printed condition (ii), which corresponds to κ = −1/2 and cannot satisfy 0 < ψ < κ for any positive ψ. Moreover, as the proof stands, Method 1 chooses R_i proportional to √t_i, not to t_i^κ, so for 0 < κ < 1/2 the stated proof would still fail unless R_i and the inclusions in (2.1) are adapted. The intended and actually used assumption appears to be κ = 1/2, i.e. inj(g(t)) ≥ β√t. The remark should be rewritten to match the corrected theorem and proof.
[Corollaries 1.1–1.3] The applications all invoke Theorem 1.1 with the stronger bound inj(g(t)) ≥ const·√t, which is exactly the bound the proofs require. Thus the applications survive the correction of Theorem 1.1. However, as the paper stands, the final steps 'the result follows from Theorem 1.1' inherit the overclaim. This is not an independent flaw, but it should be checked that the corrected Theorem 1.1 is quoted consistently throughout.

minor comments (3)

[Section 2, Method 1] The notation E_t = exp_t^{-1} is used as if exp_t were globally invertible, whereas on a complete non-compact manifold exp_t is surjective but not injective. The argument only needs E_t to be a local inverse on balls of radius below inj(g(t)); this should be stated explicitly.
[Section 2, Method 2] The set M' = ∪_{t≥0} ∂B_{g(t)}(x0, β/2√t) is claimed to be a smooth manifold 'by the injectivity condition (ii)'. At t = 0 this set is the single point (x0,0), not a sphere; smoothness near the tip t = 0 deserves a brief justification (e.g. writing the hypersurface locally as (v, t) = (v, c|v|^2)).
[Throughout] There are several minor typographical issues: 'PIC1' should presumably be 'PIC_1'; 'bi-holomorphic' should be 'biholomorphic'; and the fraction in condition (i) of Theorem 1.1 is occasionally rendered without the slash in the extracted text. These do not affect the mathematics.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the central theorem is derived from standard geometric-analysis estimates, and the cited prior results are external published theorems rather than self-referential inputs.

full rationale

The paper's central claim, Theorem 1.1, is a diffeomorphism criterion proved directly from Ricci-flow estimates: the curvature lower bound Ric(g(t)) ≥ -ψ/t, the injectivity-radius lower bound, and the geodesic-distance distortion estimate (2.2). The proof does not fit any parameter, rename an output as an input, or invoke a uniqueness theorem to force its conclusion. The applications do cite prior published long-time existence results, including [3] (Chan, Huang, Lee), which shares the first author, but that citation is used only to obtain a long-time Ricci flow with curvature and injectivity-radius bounds in Corollary 1.1, not to import the diffeomorphism conclusion itself. Such a published, externally verifiable existence result is legitimate independent support and does not make the derivation circular. The apparent gap between the printed hypothesis inj(g(t)) ≥ β/√t and the proof's actual use of radii proportional to β√t (e.g., R_i ≈ β√t_i in Method 1 and |v| ≤ (3β/4)√t in Method 2) is a real statement–proof mismatch and a correctness concern, but it is not a circularity: the proof does not reduce the target theorem to an equivalent assumption by construction. No self-definitional step, fitted-input-as-prediction step, or ansatz-smuggled-via-citation step was found.

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

The theorem itself is a self-contained geometric-analysis argument; the main external inputs are prior flow-existence/smoothing theorems used in the applications, and standard tools (geodesic length comparison, exponential map, ODE flows).

assumptions (5)

standard math Ricci flow geodesic length comparison: for Ric ≥ -ψ/t, d_{g(t')}(x,x0)/d_{g(t)}(x,x0) ≤ (t'/t)^ψ.
Used to establish ball inclusions (2.1) and the limits in Method 2. Follows from the first variation formula along minimizing geodesics.
standard math Exponential map exp_{x0,g(t)} is a diffeomorphism on Euclidean balls of radius less than the injectivity radius.
Used throughout Section 2 to define E_t and the space-time exponential map; standard Riemannian geometry.
domain assumption Existence of a complete long-time Ricci flow with bounds |Rm| ≤ C/t and inj ≥ C^{-1}√t for small-curvature-concentration initial data (from [3] and [17]).
Used in Corollary 1.1 to reduce to Theorem 1.1; these are published external results, one by the present first author.
domain assumption Short-time Ricci flow existence with local curvature/injectivity estimates for 3D non-negative Ricci with maximal volume growth (Simon-Topping [22] and Chen-Xu-Zhang [9]).
Used in Corollary 1.2 to construct a sequence of flows and pass to the limit.
domain assumption Long-time Ricci flow existence and injectivity lower bound under Sobolev and small L^{n/2} curvature conditions ([4], [7], [18]).
Used in Corollary 1.3.

how reviews work

0 comments

Cite this review

Pith. "Pith review of A note on a diffeomorphism criterion via long-time Ricci flow." pith.science (2026). https://pith.science/paper/FF6LJZU6

@misc{pith2026250905802,
  author       = {Pith},
  title        = {Pith review of: A note on a diffeomorphism criterion via long-time Ricci flow},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FF6LJZU6}},
  note         = {Machine review of arXiv:2509.05802}
}

abstract

In this note, we give a diffeomorphism (to $\mathbb{R}^n$) criterion via long-time Ricci flow and show some applications. In particular, we provide an affirmative answer that the conclusion in [Manifolds with small curvature concentration, Ann. PDE, 2024] by Chan, Lee and the first named author and [Removing scalar curvature assumption for Ricci flow smoothing, Bull. Lond. Math. Soc., 2025] by A. Martens about manifolds with small curvature concentration can be improved to diffeomorphism in dimension $4$.

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

Local mollification of metrics with small curvature concentration
math.DG 2025-10 conditional novelty 7.0 of 10

Metrics with small scale-invariant curvature concentration can be locally smoothed by Ricci flow using only Sobolev and volume-growth controls, yielding compactness and Euclidean-diffeomorphism results.

Reference graph

Works this paper leans on

24 extracted references · 18 canonical work pages · cited by 1 Pith paper

[3]

PDE10(2024), no

Pak-Yeung Chan, Shaochuang Huang, and Man-Chun Lee,Manifolds with small curva- ture concentration, Ann. PDE10(2024), no. 2, Paper No. 23, 31. MR 4804227

work page 2024
[17]

Adam Martens,Removing scalar curvature assumption for Ricci flow smoothing, Bull. Lond. Math. Soc.57(2025), no. 7, 1968–1989. MR 4936721

work page 2025
[22]

Topping,Local mollification of Riemannian metrics using Ricci flow, and Ricci limit spaces, Geom

Miles Simon and Peter M. Topping,Local mollification of Riemannian metrics using Ricci flow, and Ricci limit spaces, Geom. Topol.25(2021), no. 2, 913–948. MR 4251438

work page 2021
[4]

Albert Chau and Adam Martens,Long-time Ricci flow existence and topological rigidity from manifolds with pinched scale-invariant integral curvature, 2024, arXiv:2403.02564

work page Pith review arXiv 2024
[7]

Differential Geometry17(1982), no

Jeff Cheeger, Mikhail Gromov, and Michael Taylor,Finite propagation speed, kernel es- timates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Differential Geometry17(1982), no. 1, 15–53. MR 658471

work page 1982
[1]

Christoph B¨ ohm and Burkhard Wilking,Manifolds with positive curvature operators are space forms, Ann. of Math. (2)167(2008), no. 3, 1079–1097. MR 2415394

2008
[2]

Simon Brendle and Richard Schoen,Manifolds with 1/4-pinched curvature are space forms, J. Amer. Math. Soc.22(2009), no. 1, 287–307. MR 2449060

work page 2009
[5]

Differential Geom.73(2006), no

Albert Chau and Luen-Fai Tam,On the complex structure of K¨ ahler manifolds with nonnegative curvature, J. Differential Geom.73(2006), no. 3, 491–530. MR 2228320

work page 2006

Show all 24 references

[6]

Colding,On the structure of spaces with Ricci curvature bounded below

Jeff Cheeger and Tobias H. Colding,On the structure of spaces with Ricci curvature bounded below. I, J. Differential Geom.46(1997), no. 3, 406–480. MR 1484888

1997
[8]

Differential Geom.67(2004), no

Bing-Long Chen, Siu-Hung Tang, and Xi-Ping Zhu,A uniformization theorem for com- plete non-compact K¨ ahler surfaces with positive bisectional curvature, J. Differential Geom.67(2004), no. 3, 519–570. MR 2153028

2004
[9]

Bing-Long Chen, Guoyi Xu, and Zhuhong Zhang,Local pinching estimates in 3-dim Ricci flow, Math. Res. Lett.20(2013), no. 5, 845–855. MR 3207356

2013
[10]

Ann.327(2003), no

Bing-Long Chen and Xi-Ping Zhu,On complete noncompact K¨ ahler manifolds with positive bisectional curvature, Math. Ann.327(2003), no. 1, 1–23. MR 2005119

2003
[11]

Eric Chen, Guofang Wei, and Rugang Ye,Ricci flow and Gromov almost flat manifolds, 2022, arXiv:2203.05107

2022 arXiv
[12]

Hamilton,Three-manifolds with positive Ricci curvature, J

Richard S. Hamilton,Three-manifolds with positive Ricci curvature, J. Differential Ge- ometry17(1982), no. 2, 255–306. MR 664497

1982
[13]

Fei He and Man-Chun Lee,Weakly PIC1 manifolds with maximal volume growth, J. Geom. Anal.31(2021), no. 11, 10868–10885. MR 4310158

2021
[14]

Shaochuang Huang and Luen-Fai Tam,K¨ ahler-Ricci flow with unbounded curvature, Amer. J. Math.140(2018), no. 1, 189–220. MR 3749193

2018
[15]

Differential Geom.115(2020), no

Man-Chun Lee and Luen-Fai Tam,Chern-Ricci flows on noncompact complex manifolds, J. Differential Geom.115(2020), no. 3, 529–564. MR 4120818 A NOTE ON A DIFFEOMORPHISM CRITERION VIA LONG-TIME RICCI FLOW 9

2020
[16]

Math.193(2013), no

Gang Liu,3-manifolds with nonnegative Ricci curvature, Invent. Math.193(2013), no. 2, 367–375. MR 3090181

2013
[18]

,Sharpening a gap theorem: nonnegative Ricci and small curvature concen- tration, Calc. Var. Partial Differential Equations64(2025), no. 3, Paper No. 79, 30. MR 4861062

2025
[19]

Lei Ni and Luen-Fai Tam,K¨ ahler-Ricci flow and the Poincar´ e-Lelong equation, Com- mun. Anal. Geom.12(2004), no. 1-2, 111–141. MR 2074873

2004
[20]

Grisha Perelman,The entropy formula for the Ricci flow and its geometric applications, 2002, arXiv:math/0211159

2002 arXiv
[21]

MR 2638857

Wan-Xiong Shi,Ricci deformation of the metric on complete noncompact K¨ ahler man- ifolds, ProQuest LLC, Ann Arbor, MI, 1990, Thesis (Ph.D.)–Harvard University. MR 2638857

1990
[23]

Differential Geom

,Local control on the geometry in 3D Ricci flow, J. Differential Geom. 122 (2022), no. 3, 467–518122(2022), no. 3, 467–518. MR 4544560

2022
[24]

(Shaochuang Huang)College of Science, Sun Yat-sen University, Shenzhen, Guangdong, China

Bing Wang,The local entropy along Ricci flow—Part B: the pseudo-locality theorems, 2020, arXiv:2010.09981. (Shaochuang Huang)College of Science, Sun Yat-sen University, Shenzhen, Guangdong, China. Email address:huangshch23@mail.sysu.edu.cn (Zhuo Peng)College of Science, Sun Ya...

2020 arXiv

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [3]

PDE10(2024), no

Pak-Yeung Chan, Shaochuang Huang, and Man-Chun Lee,Manifolds with small curva- ture concentration, Ann. PDE10(2024), no. 2, Paper No. 23, 31. MR 4804227

work page 2024

[2] [17]

Adam Martens,Removing scalar curvature assumption for Ricci flow smoothing, Bull. Lond. Math. Soc.57(2025), no. 7, 1968–1989. MR 4936721

work page 2025

[3] [22]

Topping,Local mollification of Riemannian metrics using Ricci flow, and Ricci limit spaces, Geom

Miles Simon and Peter M. Topping,Local mollification of Riemannian metrics using Ricci flow, and Ricci limit spaces, Geom. Topol.25(2021), no. 2, 913–948. MR 4251438

work page 2021

[4] [4]

Albert Chau and Adam Martens,Long-time Ricci flow existence and topological rigidity from manifolds with pinched scale-invariant integral curvature, 2024, arXiv:2403.02564

work page Pith review arXiv 2024

[5] [7]

Differential Geometry17(1982), no

Jeff Cheeger, Mikhail Gromov, and Michael Taylor,Finite propagation speed, kernel es- timates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Differential Geometry17(1982), no. 1, 15–53. MR 658471

work page 1982

[6] [1]

Christoph B¨ ohm and Burkhard Wilking,Manifolds with positive curvature operators are space forms, Ann. of Math. (2)167(2008), no. 3, 1079–1097. MR 2415394

2008

[7] [2]

Simon Brendle and Richard Schoen,Manifolds with 1/4-pinched curvature are space forms, J. Amer. Math. Soc.22(2009), no. 1, 287–307. MR 2449060

work page 2009

[8] [5]

Differential Geom.73(2006), no

Albert Chau and Luen-Fai Tam,On the complex structure of K¨ ahler manifolds with nonnegative curvature, J. Differential Geom.73(2006), no. 3, 491–530. MR 2228320

work page 2006