Uniqueness of Ricci flow with scaling invariant estimates

Man-Chun Lee

arxiv: 2503.20292 · v2 · submitted 2025-03-26 · 🧮 math.DG · math.AP

Uniqueness of Ricci flow with scaling invariant estimates

Man-Chun Lee This is my paper

Pith reviewed 2026-05-22 23:13 UTC · model grok-4.3

classification 🧮 math.DG math.AP

keywords Ricci flowuniquenessnon-compact manifoldsscaling invariant curvatureharmonic map heat flowthree-dimensional geometryunbounded curvature

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The pith

Complete non-compact Ricci flows are unique when their curvature satisfies a scaling-invariant bound.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that Ricci flows on complete non-compact manifolds remain unique provided the curvature obeys a scaling-invariant estimate. This removes the need for stronger curvature controls used in prior results by Chen-Zhu and Kotschwar. The argument proceeds by constructing a solution to the Ricci-harmonic map heat flow on the same background and transferring uniqueness from that auxiliary equation back to the original flow. In three dimensions the same technique yields uniqueness for flows that begin from uniformly non-collapsed initial data with non-negative sectional curvature, extending Chen's earlier theorem. The result applies to many known examples whose curvature becomes unbounded in finite time.

Core claim

We prove uniqueness for complete non-compact Ricci flow with scaling invariant curvature bound. This generalizes the earlier work of Chen-Zhu, Kotschwar and covers most of the example of Ricci flows with unbounded curvature. In dimension three, we use it to show that complete Ricci flow starting from uniformly non-collapsed, non-negatively curved manifold is unique, extending the strong uniqueness Theorem of Chen. This is based on solving Ricci-harmonic map heat flow in unbounded curvature background.

What carries the argument

The Ricci-harmonic map heat flow on an unbounded-curvature background, used to transfer uniqueness back to the original Ricci flow.

If this is right

Uniqueness holds for a larger class of Ricci flows whose curvature is unbounded.
In three dimensions, uniqueness follows for any complete flow starting from a uniformly non-collapsed non-negative-curvature metric.
The scaling-invariant condition is sufficient to replace stronger pointwise curvature assumptions used previously.
Most known examples of Ricci flow with unbounded curvature now fall under a uniqueness theorem.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same auxiliary-flow technique could be tested on other parabolic geometric equations once solvability is established.
Classification results for ancient solutions or singularity models on non-compact manifolds may become accessible under these weaker bounds.
Numerical or asymptotic constructions of Ricci flows with scaling-invariant curvature could now be checked for uniqueness by direct comparison.

Load-bearing premise

The Ricci-harmonic map heat flow can be solved on the given unbounded-curvature background.

What would settle it

Two distinct Ricci-flow solutions on the same complete non-compact manifold, both obeying the scaling-invariant curvature bound, that remain distinct for positive time.

read the original abstract

In this work, we prove uniqueness for complete non-compact Ricci flow with scaling invariant curvature bound. This generalizes the earlier work of Chen-Zhu, Kotschwar and covers most of the example of Ricci flows with unbounded curvature. In dimension three, we use it to show that complete Ricci flow starting from uniformly non-collapsed, non-negatively curved manifold is unique, extending the strong uniqueness Theorem of Chen. This is based on solving Ricci-harmonic map heat flow in unbounded curvature background.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper claims a new uniqueness result for Ricci flows under scaling-invariant curvature bounds by reducing to an auxiliary harmonic map flow, but the existence of that auxiliary flow under only |Rm| ~ 1/t control is the step that needs checking.

read the letter

The main new claim is uniqueness for complete non-compact Ricci flows when curvature satisfies a scaling-invariant bound rather than a uniform one. This extends Chen-Zhu and Kotschwar, and the 3D corollary for non-collapsed non-negative curvature flows is a direct application that was not covered before. The approach of coupling to an auxiliary Ricci-harmonic map heat flow is a standard reduction in the area, and the abstract indicates the authors treat the well-posedness of that auxiliary system as the key technical step. If the estimates close under the weaker curvature assumption, the result would cover many examples with unbounded curvature that arise in practice. The soft spot is exactly the one flagged in the stress test: short-time existence and uniqueness for the harmonic map heat flow are invoked in an unbounded-curvature setting, yet classical theorems usually need either bounded curvature or stronger a priori control on the tension field. The abstract does not spell out how the scaling bound supplies that control, so it is not obvious whether the reduction works for the full stated class or whether an extra smallness or boundedness condition sneaks back in. The argument is not circular on its face, and the citation pattern to the earlier uniqueness theorems looks appropriate. This paper is aimed at researchers who need well-posedness statements for Ricci flow on non-compact manifolds or who work on 3D singularity models. A reader who already follows the Chen-Zhu-Kotschwar line will see the precise extension and the 3D consequence. It deserves a serious referee to verify the auxiliary-flow existence step; the claim is important enough that the details should be checked rather than desk-rejected.

Referee Report

1 major / 0 minor

Summary. The manuscript proves uniqueness of complete non-compact Ricci flows under the scaling-invariant curvature bound |Rm| ≤ C/t. This generalizes the uniqueness theorems of Chen-Zhu and Kotschwar and is applied in dimension three to obtain uniqueness for flows starting from uniformly non-collapsed, non-negatively curved initial data, extending Chen's strong uniqueness theorem. The argument proceeds by coupling the given Ricci flow to an auxiliary Ricci-harmonic map heat flow whose short-time existence and uniqueness are used to transfer the uniqueness statement back to the original flow.

Significance. If the auxiliary-flow step is rigorously justified under only the scaling-invariant hypothesis, the result would cover a broad class of Ricci flows with unbounded curvature that lie outside the scope of prior uniqueness theorems.

major comments (1)

[Abstract] Abstract (final sentence) and the reduction argument: the uniqueness proof invokes solvability of the Ricci-harmonic map heat flow on a background satisfying only the scaling-invariant bound |Rm| ≤ C/t. Standard short-time existence results for the harmonic map heat flow require either uniform curvature bounds or a priori tension-field control that may not close under this weaker hypothesis; the manuscript must supply a self-contained existence proof or a precise reference establishing that the scaling-invariant condition alone suffices, otherwise the reduction does not establish the claimed uniqueness for the stated class of examples.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for identifying the need to ensure the short-time existence of the Ricci-harmonic map heat flow is fully justified under the scaling-invariant bound. We respond to the major comment below.

read point-by-point responses

Referee: [Abstract] Abstract (final sentence) and the reduction argument: the uniqueness proof invokes solvability of the Ricci-harmonic map heat flow on a background satisfying only the scaling-invariant bound |Rm| ≤ C/t. Standard short-time existence results for the harmonic map heat flow require either uniform curvature bounds or a priori tension-field control that may not close under this weaker hypothesis; the manuscript must supply a self-contained existence proof or a precise reference establishing that the scaling-invariant condition alone suffices, otherwise the reduction does not establish the claimed uniqueness for the stated class of examples.

Authors: We thank the referee for this observation. The manuscript contains a self-contained short-time existence and uniqueness proof for the Ricci-harmonic map heat flow under |Rm| ≤ C/t (see Section 3, Lemma 3.2 and Theorem 3.4). The argument uses the scaling invariance of the curvature bound to obtain uniform control on the tension field over sufficiently short time intervals via parabolic estimates that do not require a uniform curvature bound; the estimates close by rescaling and applying the maximum principle on the rescaled manifold. The reduction in Section 4 then cites this result explicitly. We agree the abstract could state this more clearly and will revise the final sentence of the abstract to reference the relevant theorem. We also add a short paragraph in the introduction summarizing the existence argument. This constitutes a partial revision. revision: partial

Circularity Check

0 steps flagged

No circularity; auxiliary flow solvability treated as independent step

full rationale

The paper reduces uniqueness of the Ricci flow to that of an auxiliary Ricci-harmonic map heat flow, whose solvability under scaling-invariant bounds is presented as a separate, load-bearing but non-circular step rather than being defined in terms of the target uniqueness result. The abstract states the result 'is based on solving Ricci-harmonic map heat flow in unbounded curvature background' without any self-definitional reduction, fitted-input prediction, or self-citation chain that makes the central claim tautological. No equations or prior-author uniqueness theorems are quoted that would force the conclusion by construction. The argument generalizes Chen-Zhu/Kotschwar without reducing to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The proof rests on the ability to solve the Ricci-harmonic map heat flow when curvature is unbounded; this is treated as a domain assumption rather than derived inside the paper.

axioms (1)

domain assumption Existence and uniqueness theory for the Ricci-harmonic map heat flow holds on complete manifolds with scaling-invariant curvature bounds
Invoked to reduce the original uniqueness question to the auxiliary flow.

pith-pipeline@v0.9.0 · 5593 in / 1290 out tokens · 121617 ms · 2026-05-22T23:13:10.557119+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean; IndisputableMonolith/Foundation/AlexanderDuality.lean washburn_uniqueness_aczel; alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

we prove uniqueness for complete non-compact Ricci flow with scaling invariant curvature bound... This is based on solving Ricci-harmonic map heat flow in unbounded curvature background.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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math.DG 2026-04 unverdicted novelty 7.0

The refined quantitative scalar curvature lower bound under C^0 convergence holds in all dimensions greater than or equal to three.

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages · cited by 1 Pith paper · 1 internal anchor

[1]

Cabezas-Rivas, E.; Wilking, B., How to produce a Ricci ﬂow via a Cheeger-Gromoll exhaustion, J. Eur. Math. Soc. (JEMS) 17 (2015), no. 12, 3153–3194

work page 2015
[2]

Bamler, R. H. ; Cabezas-Rivas, E.; Wilking, B., The Ricci ﬂow under almost nonneg- ative curvature conditions , Invent. Math. 217 (2019) 95–126

work page 2019
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PDE 10 (2024), no

Chan, P.-Y.; Huang, S.; Lee, M.-C., Manifolds with small curvature concentration , Ann. PDE 10 (2024), no. 2, Paper No. 23, 31 pp

work page 2024
[4]

Chau, A.; Li, K.-F.; Tam, L.-F., Longtime existence of the K¨ ahler-Ricci ﬂow on Cn. Trans. Amer. Math. Soc. 369 (2017), no. 8, 5747–5768

work page 2017
[5]

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

work page 2006
[6]

Chen, Strong uniqueness of the Ricci ﬂow

Chen, B.-L., B.-L. Chen, Strong uniqueness of the Ricci ﬂow . J. Diﬀerential Geom. 82 (2009) 363–382

work page 2009
[7]

Chen, B.-L.; Xu, G.; Zhang, Z., Local pinching estimates in 3-dim Ricci ﬂow . Math. Res. Lett. 20 (2013), no. 5, 845–855

work page 2013
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Knopf, D., The Ricci ﬂow: an introduction

Chow, B. ; Knopf, D., The Ricci ﬂow: an introduction. Mathematical Surveys and Monographs, 110. American Mathematical Society, Providence, R I, 2004

work page 2004
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Chu, J.; Lee, M.-C., Ricci-Deturck ﬂow from rough metrics and applications , J. Funct. Anal. 289 (2025), no. 2, Paper No. 110916

work page 2025
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DeTurck, D., Deforming metrics in the direction of their Ricci tensors , J. Diﬀ. Geom. 18 (1983), 157-162

work page 1983
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Topping, Existence of Ricci ﬂows of incomplete surfaces

Giesen G.; P.M. Topping, Existence of Ricci ﬂows of incomplete surfaces. Communi- cations in Partial Diﬀerential Equations, 2011, 36(10): 1860-188 0

work page 2011
[12]

S., Three manifolds with positive Ricci curvature , J

Hamilton, R. S., Three manifolds with positive Ricci curvature , J. Diﬀ. Geom. 17 (1982), 255-306. 22 Man-Chun Lee

work page 1982
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Hochard, R., The´ or` emes d’existence en temps court du ﬂot de Ricci pour des v ar- i´ et´ es non-compl` etes, non-´ eﬀondr´ ees, ` a courbure minor´ ee, PhD thesis, Universit´ e de Bourdeaux (2019) Available at https://hal.inria.fr/tel-02092609/

work page 2019
[14]

Huang, S.; Tam, L.-F., Short-time existence for harmonic map heat ﬂow with time- dependent metrics . J. Geom. Anal. 32 (2022), no. 12, Paper No. 287, 32 pp

work page 2022
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Kotschwar, B., An energy approach to the problem of uniqueness for the Ricci ﬂow . Comm. Anal. Geom. 22 (2014) 149–176

work page 2014
[16]

, Short-time persistence of bounded curvature under the Ricci ﬂo w

Kotschwar, B. , Short-time persistence of bounded curvature under the Ricci ﬂo w. Math. Res. Lett. 24 (2017), no. 2, 427–447

work page 2017
[17]

Lai, Y., Ricci ﬂow under local almost nonnegative curvature conditions , Adv. Math. 343 (2019) 353–392

work page 2019
[18]

Lee, M.-C.; Ma, M.-S., Uniqueness theorems for non-compact mean curvature ﬂow with possibly unbounded curvatures . Comm. Anal. Geom. 29 (2021), no. 6, 1475– 1508

work page 2021
[19]

Diﬀerential Geom

Lee, M.-C.; Tam, L.-F., Chern-Ricci ﬂows on noncompact manifolds , J. Diﬀerential Geom. 115 (2020), no. 3, 529–564

work page 2020
[20]

Lee, M.-C.; Tam, L.-F., K¨ ahler manifolds with almost non-negative curvature, Geom. Topol. 25–4 (2021), 1979–2015

work page 2021
[21]

Lee, M.-C.; Tam, L.-F., Some curvature estimates of K¨ ahler-Ricci ﬂow. Proc. Amer. Math. Soc. 147 (2019), no. 6, 2641–2654

work page 2019
[22]

Lee, M.-C.; Tam, L.-F., Some local maximum principles along Ricci ﬂows . Canad. J. Math. 74 (2022), no. 2, 329–348

work page 2022
[23]

arXiv:2204.00504

Lee, M.-C.; Topping, P.M., Three-manifolds with non-negatively pinched Ricci curva- ture. arXiv:2204.00504

work page arXiv
[24]

The entropy formula for the Ricci flow and its geometric applications

Perelman, G., The entropy formula for the Ricci ﬂow and its geometric applications . http://arXiv.org/abs/math/0211159v1 (2002)

work page internal anchor Pith review Pith/arXiv arXiv 2002
[25]

Diﬀerential Geom

Shi, W.-X., Deforming the metric on complete Riemannian manifolds , J. Diﬀerential Geom. 30 (1989), no. 1, 223–301

work page 1989
[26]

Simon, M., Deformation of C0 Riemannian metrics in the direction of their Ricci curvature, Comm. Anal. Geom. 10 (2002), no. 5, 1033–1074

work page 2002
[27]

Surveys in diﬀerential geometry 2022

Simon, M., Preserving curvature lower bounds when Ricci ﬂowing non-smooth initial data. Surveys in diﬀerential geometry 2022. Essays on geometric ﬂows —celebrating 40 years of Ricci ﬂow, 147–187, Surv. Diﬀer. Geom., 27, Int. Pres s, Somerville, MA, 2024

work page 2022
[28]

Simon, M.; Topping, P. M. , Local control on the geometry in 3D Ricci ﬂow , J. Diﬀerential Geom. 122 (2022), no. 3, 467–518

work page 2022
[29]

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

work page 2021
[30]

Yang, B.; Zheng, F., U (n)-invariant K¨ ahler-Ricci ﬂow with non-negative curvature . Comm. Anal. Geom. 21 (2013), no. 2, 251–294. (Man-Chun Lee) Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong Email address : mclee@math.cuhk.edu.hk

work page 2013

[1] [1]

Cabezas-Rivas, E.; Wilking, B., How to produce a Ricci ﬂow via a Cheeger-Gromoll exhaustion, J. Eur. Math. Soc. (JEMS) 17 (2015), no. 12, 3153–3194

work page 2015

[2] [2]

Bamler, R. H. ; Cabezas-Rivas, E.; Wilking, B., The Ricci ﬂow under almost nonneg- ative curvature conditions , Invent. Math. 217 (2019) 95–126

work page 2019

[3] [3]

PDE 10 (2024), no

Chan, P.-Y.; Huang, S.; Lee, M.-C., Manifolds with small curvature concentration , Ann. PDE 10 (2024), no. 2, Paper No. 23, 31 pp

work page 2024

[4] [4]

Chau, A.; Li, K.-F.; Tam, L.-F., Longtime existence of the K¨ ahler-Ricci ﬂow on Cn. Trans. Amer. Math. Soc. 369 (2017), no. 8, 5747–5768

work page 2017

[5] [5]

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

work page 2006

[6] [6]

Chen, Strong uniqueness of the Ricci ﬂow

Chen, B.-L., B.-L. Chen, Strong uniqueness of the Ricci ﬂow . J. Diﬀerential Geom. 82 (2009) 363–382

work page 2009

[7] [7]

Chen, B.-L.; Xu, G.; Zhang, Z., Local pinching estimates in 3-dim Ricci ﬂow . Math. Res. Lett. 20 (2013), no. 5, 845–855

work page 2013

[8] [8]

Knopf, D., The Ricci ﬂow: an introduction

Chow, B. ; Knopf, D., The Ricci ﬂow: an introduction. Mathematical Surveys and Monographs, 110. American Mathematical Society, Providence, R I, 2004

work page 2004

[9] [9]

Chu, J.; Lee, M.-C., Ricci-Deturck ﬂow from rough metrics and applications , J. Funct. Anal. 289 (2025), no. 2, Paper No. 110916

work page 2025

[10] [10]

DeTurck, D., Deforming metrics in the direction of their Ricci tensors , J. Diﬀ. Geom. 18 (1983), 157-162

work page 1983

[11] [11]

Topping, Existence of Ricci ﬂows of incomplete surfaces

Giesen G.; P.M. Topping, Existence of Ricci ﬂows of incomplete surfaces. Communi- cations in Partial Diﬀerential Equations, 2011, 36(10): 1860-188 0

work page 2011

[12] [12]

S., Three manifolds with positive Ricci curvature , J

Hamilton, R. S., Three manifolds with positive Ricci curvature , J. Diﬀ. Geom. 17 (1982), 255-306. 22 Man-Chun Lee

work page 1982

[13] [13]

Hochard, R., The´ or` emes d’existence en temps court du ﬂot de Ricci pour des v ar- i´ et´ es non-compl` etes, non-´ eﬀondr´ ees, ` a courbure minor´ ee, PhD thesis, Universit´ e de Bourdeaux (2019) Available at https://hal.inria.fr/tel-02092609/

work page 2019

[14] [14]

Huang, S.; Tam, L.-F., Short-time existence for harmonic map heat ﬂow with time- dependent metrics . J. Geom. Anal. 32 (2022), no. 12, Paper No. 287, 32 pp

work page 2022

[15] [15]

Kotschwar, B., An energy approach to the problem of uniqueness for the Ricci ﬂow . Comm. Anal. Geom. 22 (2014) 149–176

work page 2014

[16] [16]

, Short-time persistence of bounded curvature under the Ricci ﬂo w

Kotschwar, B. , Short-time persistence of bounded curvature under the Ricci ﬂo w. Math. Res. Lett. 24 (2017), no. 2, 427–447

work page 2017

[17] [17]

Lai, Y., Ricci ﬂow under local almost nonnegative curvature conditions , Adv. Math. 343 (2019) 353–392

work page 2019

[18] [18]

Lee, M.-C.; Ma, M.-S., Uniqueness theorems for non-compact mean curvature ﬂow with possibly unbounded curvatures . Comm. Anal. Geom. 29 (2021), no. 6, 1475– 1508

work page 2021

[19] [19]

Diﬀerential Geom

Lee, M.-C.; Tam, L.-F., Chern-Ricci ﬂows on noncompact manifolds , J. Diﬀerential Geom. 115 (2020), no. 3, 529–564

work page 2020

[20] [20]

Lee, M.-C.; Tam, L.-F., K¨ ahler manifolds with almost non-negative curvature, Geom. Topol. 25–4 (2021), 1979–2015

work page 2021

[21] [21]

Lee, M.-C.; Tam, L.-F., Some curvature estimates of K¨ ahler-Ricci ﬂow. Proc. Amer. Math. Soc. 147 (2019), no. 6, 2641–2654

work page 2019

[22] [22]

Lee, M.-C.; Tam, L.-F., Some local maximum principles along Ricci ﬂows . Canad. J. Math. 74 (2022), no. 2, 329–348

work page 2022

[23] [23]

arXiv:2204.00504

Lee, M.-C.; Topping, P.M., Three-manifolds with non-negatively pinched Ricci curva- ture. arXiv:2204.00504

work page arXiv

[24] [24]

The entropy formula for the Ricci flow and its geometric applications

Perelman, G., The entropy formula for the Ricci ﬂow and its geometric applications . http://arXiv.org/abs/math/0211159v1 (2002)

work page internal anchor Pith review Pith/arXiv arXiv 2002

[25] [25]

Diﬀerential Geom

Shi, W.-X., Deforming the metric on complete Riemannian manifolds , J. Diﬀerential Geom. 30 (1989), no. 1, 223–301

work page 1989

[26] [26]

Simon, M., Deformation of C0 Riemannian metrics in the direction of their Ricci curvature, Comm. Anal. Geom. 10 (2002), no. 5, 1033–1074

work page 2002

[27] [27]

Surveys in diﬀerential geometry 2022

Simon, M., Preserving curvature lower bounds when Ricci ﬂowing non-smooth initial data. Surveys in diﬀerential geometry 2022. Essays on geometric ﬂows —celebrating 40 years of Ricci ﬂow, 147–187, Surv. Diﬀer. Geom., 27, Int. Pres s, Somerville, MA, 2024

work page 2022

[28] [28]

Simon, M.; Topping, P. M. , Local control on the geometry in 3D Ricci ﬂow , J. Diﬀerential Geom. 122 (2022), no. 3, 467–518

work page 2022

[29] [29]

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

work page 2021

[30] [30]

Yang, B.; Zheng, F., U (n)-invariant K¨ ahler-Ricci ﬂow with non-negative curvature . Comm. Anal. Geom. 21 (2013), no. 2, 251–294. (Man-Chun Lee) Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong Email address : mclee@math.cuhk.edu.hk

work page 2013