Type-I Blowup Solutions for Yang-Mills Flow

Jaehwan Kim; Sanghoon Lee

REVIEW 2 major objections 5 minor 60 references

Type-I Blowup Solutions for Yang-Mills Flow

T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper constructs an infinite-dimensional family of Yang-Mills flow solutions that converge to the equivariant shrinking soliton up to gauge, without imposing any symmetry on the perturbation.

desk verdict Non-equivariant Type-I blowups are the right target and the strategy is standard, but the conjugated operator in (2.5) is written inconsistently and the spectral lemma built on it is load-bearing; needs a fix before the main theorem can be trusted. read the letter →

arxiv 2411.19293 v1 pith:4U76FEFL submitted 2024-11-28 math.DG

Jaehwan Kim , Sanghoon Lee This is my paper · ORCID

classification math.DG MSC 53C4458E1535B44

keywords Yang-MillsflowType-Iblowuphomotheticallyshrinkingsolitonde-Turckspectralstabilitynon-equivariantsolutionsGaussianweightedspacesKatoinequality

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 the Yang-Mills flow on $\mathbb{R}^n$ with structure group $SO(n)$, for $5 \leq n \leq 9$, admits an infinite-dimensional family of solutions that blow up at a finite time and converge, modulo gauge transformations, to the known $SO(n)$-equivariant homothetically shrinking soliton $W$. This is the first construction of such converging solutions without imposing $SO(n)$-equivariance on the perturbation, so stability of $W$ holds in the full space of connections. The family is parameterized by arbitrary small initial connections lying in an infinite-dimensional weighted Hölder space, subject only to a spectral projection condition that removes the finitely many non-positive modes. As a corollary, the paper constructs a non-equivariant, asymmetric Type-I blowup solution. Consequently, the stable manifold of $W$ in the gauge-fixed flow is infinite-dimensional.

What carries the argument

The load-bearing object is the spectral projection $\Pi_{>0}$ of the linearized operator $-L$ of the rescaled Yang-Mills de-Turck flow at the soliton $W$, acting on the Gaussian-weighted space $L^2_\rho$ with $\rho = e^{-|x|^2/4}$. Lemma 2.1 gives $-L$ a complete orthonormal eigenbasis with finitely many non-positive eigenvalues, so $\Pi_{>0}$ selects the modes that decay exponentially in the similarity time $\tau$. The explicit solution formula (3.4) represents the linearized solution using forward-in-time integrals for positive eigenvalues and backward integrals for non-positive ones, and a refined Kato inequality for the Frobenius norm (Lemma 3.2) supplies global sup-norm control that standard comparison arguments cannot give for tensor-valued sections. These pieces combine with weighted parabolic Hölder spaces to run the contraction-mapping argument that produces the nonlinear solution.

What would settle it

Compute the spectrum of the linearized operator $-L$ in the Gaussian $L^2$ space: if any essential spectrum reaches or crosses zero, or if the eigenfunctions fail to form a complete orthonormal basis, then the projection formula (3.4) and all subsequent contraction estimates collapse. A more targeted test: exhibit an eigenfunction with eigenvalue $\lambda > 1/2$ whose pointwise growth exceeds $C r^{2\lambda - 1}$; Lemma 3.10 asserts no such eigenfunction exists, and that growth bound is used to control the initial data and the non-equivariant solution.

Watch

Extended reading notes

Core claim

Fix $5 \leq n \leq 9$ and let $W$ be the explicit homothetically shrinking soliton found by Weinkove. For every smooth connection $\tilde A_0$ sufficiently close to $W$ in a weighted Hölder norm, there is a smooth solution $A$ of the Yang-Mills flow, defined up to time $1$, such that the spectral projection condition $\Pi_{>0}(A(\cdot,0)-W-\tilde A_0)=0$ holds and, after applying a time-dependent gauge transformation, $A$ converges to $W$ at the rate $(1-t)^\delta$ relative to the soliton's $L^\infty$ scale. The proof works with the rescaled Yang-Mills de-Turck flow, whose linearization at $W$ is a self-adjoint operator with discrete spectrum in a Gaussian $L^2_\rho$ space; projecting initial data onto the positive spectral subspace and running a fixed-point argument yields the solution. This establishes, in the paper's language, that the unstable and neutral directions can be freely chosen while the positive modes control convergence, and it yields the corollary that a non-$SO(n)$-equivariant solution exists.

Load-bearing premise

The construction assumes that near the soliton every perturbation splits cleanly into a finite number of decaying modes and an infinite number of growing modes, with no continuous spectrum in between; if that spectral picture failed, the projection that selects converging initial data would not exist.

Editorial extensions

If this is right

There exist asymmetric Type-I blowup solutions of the Yang-Mills flow, i.e., solutions whose curvature blows up with the maximal self-similar rate, that are not gauge equivalent to any $SO(n)$-equivariant solution.
The stable manifold of $W$ in the full non-equivariant, gauge-fixed Yang-Mills flow is infinite-dimensional: the admissible perturbations $\tilde A_0$ range over an infinite-dimensional weighted Hölder space.
In dimensions $5$ through $9$, the equivariant soliton $W$ is dynamically stable in the full space of connections, not just in the $SO(n)$-equivariant class, in the sense that every sufficiently small perturbation admits a smooth solution converging to $W$ up to gauge.
The convergence is exponential in similarity coordinates, giving the explicit bound $\frac{\|W(\cdot,t)-S(t)^* A(\cdot,t)\|_{L^\infty}}{\|W(\cdot,t)\|_{L^\infty}} \leq M\|\tilde A_0\| (1-t)^\delta$ for all $t \in [0,\infty)$.
The proof supplies a concrete non-symmetric initial datum by cutting off $W$ far from the origin, so the asymmetry of the resulting blowup solution appears at arbitrarily large spatial scales.

Reading between the lines

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

Editorial inference: the explicit translational eigenfunctions $F_{W\alpha}$ with eigenvalue $-1/2$ should generate a family of blowup solutions with drifting centers; tracking the coefficients of these modes would quantify the drift rate.
Editorial inference: because the lowest eigenfunction (eigenvalue $-1$) and the translational modes (eigenvalue $-1/2$) control the leading decay, generic non-equivariant solutions should approach $W$ with a small spatial translation; this is a testable prediction from the coefficients of $A(\cdot,0)-W$ on those two eigenspaces.
Editorial inference: the method is not tied to the specific structure group; it should adapt to any gauge-gradient flow whose linearized operator at a soliton has discrete Gaussian spectrum with finitely many non-positive eigenvalues, such as harmonic map flow.

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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

Short version: this is a serious stable-manifold construction, and the non-equivariant setting is genuinely new. But the paper has a load-bearing problem in Section 2.2 that needs fixing before the main theorem is established.

What is new: prior work (Donninger–Schörkhuber, Glogić–Schörkhuber) only handled SO(n)-equivariant stability and blowup. Here the authors build an infinite-dimensional family of solutions of the rescaled de-Turck flow converging to Weinkove's soliton W, without any symmetry assumption, for 5 ≤ n ≤ 9, and then use a cutoff perturbation to produce a solution not gauge-equivalent to any equivariant one. That is a real step beyond the literature. The refined Kato inequality in Lemma 3.2 and the weighted parabolic Hölder setup are sensible tools; the nonlinear fixed-point argument in Section 4 follows the standard blueprint and, conditional on the linear theory, looks coherent. The citation pattern is honest: W and the spectral background come from Weinkove and DS/GS, and the paper flags the missing non-equivariant gap.

The soft spot is real and central. In Eq (2.5) the two displayed formulas for A = -U^{-1}LU are not the same operator. Conjugating the term 2[W_i, ∂_i u_j] by U = e^{|·|^2/8} produces a y_i[W_i, φ_j] term that appears in neither display. The coefficient of [φ_i, F_Wij] changes from 2 in the first display to 1 in the second. The zeroth-order potential in the first display is 1/16(4(n-2)-|y|^2) - 1/2, while the second display has 1/16(|y|^2 - 4(n-2)); these differ by more than sign. Moreover, (D_W D_W^* + D_W^* D_W) contributes -2Δ plus a curvature term, so it cannot equal the first display's -Δ plus 2[W_i, ∂_i φ_j]. One of these expressions is a transcription error, or the operator L as written is not the one being analyzed.

This matters because Lemma 2.1 is load-bearing: it supplies the discrete L^2_ρ eigenbasis, the finite non-positive spectrum, and the projector Π_{>0} that defines the initial condition and the explicit solution formula (3.4). The proof of Lemma 2.1 runs on the second display and on self-adjointness of A. If the true conjugated operator has the extra derivative and potential terms, then the spectral decomposition used everywhere is not established. The rest of the paper cannot be certified until this is repaired.

Who this is for: people working on Yang–Mills flow singularity formation and stable-manifold constructions near solitons. The paper deserves a serious referee despite my skepticism about the current state of (2.5), because the result, if fixable, is significant and the surrounding machinery is substantive. I would not cite it or put it in a reading group until the spectral lemma is corrected.

Referee Report

2 major / 5 minor

Summary. The paper constructs, for 5 ≤ n ≤ 9, an infinite-dimensional family of smooth solutions to the Yang–Mills flow on R^n with structure group SO(n) that converge, modulo gauge, to Weinkove's homothetically shrinking SO(n)-equivariant soliton W, thereby producing asymmetric Type-I blowup solutions. The proof proceeds through the de-Turck gauge, similarity variables, linearization at W, a Gaussian L^2 spectral decomposition of the linearized operator, weighted parabolic Hölder estimates, and a fixed-point argument on a projected stable slice, followed by gauge reconstruction. The main results are Theorem 1.1, Theorem 4.2, and Corollary 1.2.

Significance. If correct, this would be a notable advance: it would give the first construction of non-equivariant Type-I blowup for the Yang–Mills flow in dimensions 5 ≤ n ≤ 9 and would show that the stable manifold of the soliton is infinite-dimensional in the full gauge-fixed problem. The overall strategy is appropriate, and the construction is not circular: the soliton is imported explicitly from Weinkove's work, the linearized equation is derived from the flow, and the contradiction argument for non-equivariance is self-contained. The explicit translation eigenfunctions and the refined Kato inequality are potentially useful contributions. The significance is contingent, however, on correcting the linearization and spectral input discussed in the major comments.

major comments (2)

[§2.2, Eq. (2.5); §2.1, Eq. (2.4)] The displayed identities defining L and A are internally inconsistent. In Eq. (2.4), the first expression for L ends with the term 2[u_i, ∂_i W_j − ∂_j W_i + [W_i, W_j]], while the second expression ends with −[u_i, F_{W,ij}] for the same bracket; these differ by a factor of −2, so the equality in (2.4) is false as written. The problem propagates to Eq. (2.5): conjugating the first expression of (2.4) by U = e^{|y|^2/8} produces, from the term 2[W_i, ∂_i u_j], an additional term (1/2)y_i[W_i, φ_j], which is absent from both displayed formulas for A. In addition, the coefficient of [φ_i, F_{W,ij}] changes from 2 in the first display of (2.5) to 1 in the second, and the quadratic potentials differ by an additive constant. Consequently, Lemma 2.1 is proved for an operator that is not shown to be the operator L that appears in Eq. (3.1).
[§2.2, Lemma 2.1] Lemma 2.1 is the load-bearing spectral input of the paper. It supplies the complete L^2_ρ eigenbasis, the finite set of nonpositive eigenvalues, and the spectral gap λ_{I+1} > 0 that are used in the explicit solution formula (3.4), the decay estimate (3.5), Lemma 3.12, and the contraction estimates in Theorem 4.1. Because the operator on which Lemma 2.1 is proved is not unambiguously the true conjugated linearization, the current manuscript does not establish this spectral input. The authors must recompute A for the actual L and prove discreteness and self-adjointness (or a suitable spectral substitute) for that operator. A bounded-perturbation repair may be possible, but it is not supplied; without it the stable-manifold construction has no foundation.

minor comments (5)

[§3.2, Lemma 3.1] The displayed Hölder manipulations in Eqs. (3.6) and (3.7) appear to contain missing squares and misplaced factors; please rewrite these lines carefully, since they are the source of the exponential decay bounds.
[§4, Theorem 4.2] The statement says the convergence estimate holds for all t ∈ [0, ∞), but the flow blows up at t = 1 and the displayed estimate contains factors (1 − t)^δ; the intended time interval is presumably [0, 1).
[§2.1, Eq. (2.4)] The notation ∇_{W_i} and the action of D_W on 1-forms should be defined before Eq. (2.4), since the second displayed expression for L uses this notation without prior explanation.
[§3.2, end of proof of Lemma 3.1] The formula for u[a]_j contains the fragment 'u_j(τ)dσ' for j = 1, ..., I; this should be completed, presumably with an integral or with the coefficient a_j.
[§2.2, Eq. (2.5)] The symbols φ and φ are used inconsistently in the displayed formulas for A; please unify the notation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the soliton, spectral theorem, and Schauder estimates are imported from external sources, and the fixed-point construction does not assume the convergence it proves.

full rationale

The paper's derivation chain is self-contained in the sense required by the circularity check. The soliton W is imported with explicit constants from Weinkove's external existence result, not defined in terms of the paper's own target solutions. The linearized operator L is obtained by direct computation from the rescaled Yang-Mills de-Turck flow, and the spectral lemma is justified by a compactness argument citing Reed and Simon, with the only 'explicit eigenfunctions' being derived from the geometric translations of the soliton. The projection condition used in the Perron-type fixed-point argument is a standard device: it selects initial data whose component in the growing eigenspace is fixed, and the exponential convergence is then proven from the spectral gap and contraction estimates, not assumed by the definition of the solution space. The non-equivariant example in Corollary 1.2 is constructed from an explicit cutoff of W, and the asymmetry is verified by a gauge-invariant curvature norm computation, so it is not a renamed known result or a fitted prediction. The paper contains no load-bearing self-citations and does not invoke a uniqueness theorem from the authors' own prior work. The skeptic's concern about the two displays in equation (2.5) possibly defining different operators would, if valid, be a correctness gap in the proof of Lemma 2.1, not a circular reduction: an unproved spectral lemma is a missing justification, not an input repackaged as an output. Accordingly, no circular step satisfies the evidentiary standard, and the appropriate circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted to data. The central claim rests on standard spectral theory, the known Weinkove soliton W, the de-Turck gauge formalism, and parabolic Schauder estimates; the paper introduces no new invented entities. The main nonstandard analytical input is the refined Kato inequality and the weighted Holder framework, both proven in the paper.

assumptions (4)

domain assumption The linearized operator -L has purely discrete spectrum with a complete orthonormal eigenbasis in the Gaussian L^2_rho space and finitely many nonpositive eigenvalues.
Proved via Reed-Simon compactness criteria in Lemma 2.1, but it is the foundation for the spectral projector Pi>0 and the exponential decay estimates; if wrong, the fixed-point construction fails.
standard math The de-Turck gauge transformation S determined by Eq (2.2) exists globally and converts the gauge-fixed solution into a genuine Yang-Mills flow solution.
Invoked through Struwe's work; standard in Yang-Mills flow theory but load-bearing for translating the rescaled de-Turck result back to the original flow.
domain assumption The soliton W of Weinkove exists for 5 <= n <= 9 and satisfies the soliton equation (1.2).
Taken from the cited literature; the restriction 5 <= n <= 9 comes from existence of W and the Bizon-Wasserman nonexistence for n >= 10.
standard math Parabolic Schauder estimates and standard weak solution theory for parabolic systems apply to the tensor-valued system with bounded coefficients on compact regions.
Used throughout Sections 3 and 4 for C^{2,alpha} regularity and bootstrapping.

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Pith. "Pith review of Type-I Blowup Solutions for Yang-Mills Flow." pith.science (2026). https://pith.science/paper/4U76FEFL

@misc{pith2026241119293,
  author       = {Pith},
  title        = {Pith review of: Type-I Blowup Solutions for Yang-Mills Flow},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4U76FEFL}},
  note         = {Machine review of arXiv:2411.19293}
}

abstract

In this paper, we construct an infinite-dimensional family of solutions for the Yang-Mills flow on $\mathbb{R}^n \times SO(n)$ for $5 \leq n \leq 9$, which converge to $SO(n)$-equivariant homothetically shrinking solitons, modulo the gauge group. As a corollary, we prove the existence of asymmetric Type-I blowup solutions for the Yang-Mills flow.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

60 extracted references · 52 canonical work pages

[1]

Angenent, S

S. Angenent, S. Brendle, P. Daskalopoulos, and N. Sesum. Unique asymptotics of compact ancient solutions to three-dimensional ricci flow. Comm. Pure Appl. Math. , 75(5):1032--1073, 2022

work page 2022
[2]

Angenent, P

S. Angenent, P. Daskalopoulos, and N. Sesum. Unique asymptotics of ancient convex mean curvature flow solutions. J. Differential Geom. , 111(3):381--455, 2019

2019
[3]

Angenent, P

S. Angenent, P. Daskalopoulos, and N. Sesum. Uniqueness of two-convex closed ancient solutions to the mean curvature flow. Ann. of Math. , 192(2):353--436, 2020

work page 2020
[4]

S. B. Angenent and J. J. L. Velazquez. Degenerate neckpinches in mean curvature flow. J. Reine Angew. Math. , 482:15--66, 1997

work page 1997
[5]

Brendle and K

S. Brendle and K. Choi. Uniqueness of convex ancient solutions to mean curvature flow in R ^3 . Invent. Math. , 217(1):35--76, 2019

work page 2019
[6]

Brendle and K

S. Brendle and K. Choi. Uniqueness of convex ancient solutions to mean curvature flow in higher dimensions. Geom. Topol. , 25(5):2195--2234, 2021

2021
[7]

Brendle, P

S. Brendle, P. Daskalopoulos, K. Naff, and N. Sesum. Uniqueness of compact ancient solutions to the higher-dimensional ricci flow. J. Reine Angew. Math. , 2023(795):85--138, 2023

work page 2023
[8]

Brendle, P

S. Brendle, P. Daskalopoulos, and N. Sesum. Uniqueness of compact ancient solutions to three-dimensional ricci flow. Invent. Math. , 226(2):579--651, 2021

work page 2021

Show all 60 references

[9]

Brendle and K

S. Brendle and K. Naff. Rotational symmetry of ancient solutions to the ricci flow in higher dimensions. Geom. Topol. , 27(1):153--226, 2023

2023
[10]

S. Brendle. Ancient solutions to the ricci flow in dimension 3. Acta Math. , 225(1):1--102, 2020

2020
[11]

Bizo\' n and A

P. Bizo\' n and A. Wasserman. Nonexistence of shrinkers for the harmonic map flow in higher dimensions. Int. Math. Res. Not. IMRN , (17):7757--7762, 2015

2015
[12]

Bernstein and L

J. Bernstein and L. Wang. A topological property of asymptotically conical self-shrinkers of small entropy. Duke Math. J. , 166(3):403--435, 2017

2017
[13]

Chodosh, K

O. Chodosh, K. Choi, C. Mantoulidis, and F. Schulze. Mean curvature flow with generic initial data. Invent. Math. , pages 1--100, 2024

2024
[14]

Carlotto, O

A. Carlotto, O. Chodosh, and Y. Rubinstein. Slowly converging yamabe flows. Geom. Topol. , 19(3):1523--1568, 2015

2015
[15]

Chodosh, K

O. Chodosh, K. Choi, and F. Schulze. Mean curvature flow with generic initial data II . arXiv preprint arXiv:2302.08409 , 2023

2023 arXiv
[16]

B. Choi, P. Daskalopoulos, W. Du, R. Haslhofer, and N. Sesum. Classification of bubble-sheet ovals in r4. arXiv preprint arXiv:2209.04931 , 2022

2022 arXiv
[17]

Choi and P.-K

B. Choi and P.-K. Hung. Thom's gradient conjecture for nonlinear evolution equations. arXiv preprint arXiv:2405.17510 , 2024

2024 arXiv
[18]

K. Choi, R. Haslhofer, and O. Hershkovits. Ancient low-entropy flows, mean-convex neighborhoods, and uniqueness. Acta Math. , 228(2):217--301, 2022

2022
[19]

K. Choi, R. Haslhofer, and O. Hershkovits. A nonexistence result for wing-like mean curvature flows in R ^4 . Geom. Topol. , 28(7):3095--3134, 2024

2024
[20]

K. Choi, R. Haslhofer, O. Hershkovits, and B. White. Ancient asymptotically cylindrical flows and applications. Invent. Math. , 229(1):139--241, 2022

2022
[21]

T. H. Colding and W. P. Minicozzi. Generic mean curvature flow I ; generic singularities. Ann. of Math. , pages 755--833, 2012

2012
[22]

Choi and C

K. Choi and C. Mantoulidis. Ancient gradient flows of elliptic functionals and morse index. Amer. J. Math. , 144(2):541--573, 2022

2022
[23]

Chodosh and F

O. Chodosh and F. Schulze. Uniqueness of asymptotically conical tangent flows. Duke Math. J. , 170(16):3601--3657, 2021

2021
[24]

Chen and Y

Z. Chen and Y. Zhang. Stabilities of homothetically shrinking yang-mills solitons. Trans. Amer. Math. Soc. , 367(7):5015--5041, 2015

2015
[25]

D. M. DeTurck. Deforming metrics in the direction of their ricci tensors. J. Differential Geom. , 18(1):157--162, 1983

1983
[26]

Du and R

W. Du and R. Haslhofer. Hearing the shape of ancient noncollapsed flows in R ^4 . Comm. Pure Appl. Math. , 77(1):543--582, 2024

2024
[27]

Daskalopoulos, R

P. Daskalopoulos, R. Hamilton, and N. Sesum. Classification of compact ancient solutions to the curve shortening flow. J. Differential Geom. , 84(3):455--464, 2010

2010
[28]

Daskalopoulos, R

P. Daskalopoulos, R. Hamilton, and N. Sesum. Classification of ancient compact solutions to the ricci flow on surfaces. J. Differential Geom. , 91(2):171--214, 2012

2012
[29]

S. K. Donaldson and P. B. Kronheimer. The geometry of four-manifolds . Oxford Univ. Press, 1997

1997
[30]

Donninger and B

R. Donninger and B. Sch \"o rkhuber. Stable blowup for the supercritical yang-mills heat flow. J. Differential Geom. , 113(1):55--94, 2019

2019
[31]

Logarithmic sobolev inequalities on submanifolds of euclidean space

Klaus Ecker. Logarithmic sobolev inequalities on submanifolds of euclidean space. J. Reine Angew. Math. , pages 105--118, 2000

2000
[32]

L. C. Evans. Partial differential equations , volume 19. Amer. Math. Soc., 2022

2022
[33]

P. Feehan. Global existence and convergence of solutions to gradient systems and applications to yang-mills gradient flow, 2014

2014
[34]

A. Gastel. Singularities of first kind in the harmonic map and yang-mills heat flows. Math. Z. , 242:47--62, 2002

2002
[35]

J. F. Grotowski. Finite time blow-up for the yang-mills heat flow in higher dimensions. Math. Z. , 237(2):321--333, 2001

2001
[36]

Glogi \'c and B

I. Glogi \'c and B. Sch \"o rkhuber. Nonlinear stability of homothetically shrinking yang-mills solitons in the equivariant case. Comm. Partial Differential Equations , 45(8):887--912, 2020

2020
[37]

Hong and G

M.-C. Hong and G. Tian. Asymptotical behavior of the yang-mills flow and singular yang-mills connections. Math. Ann. , 330:441--472, 2004

2004
[38]

Hong and G

M.-C. Hong and G. Tian. Global existence of the m-equivariant yang-mills flow in four-dimensional spaces. Comm. Anal. Geom. , 12(1):183--211, 2004

2004
[39]

B. F. Knerr. Parabolic interior schauder estimates by the maximum principle. Arch. Ration. Mech. Anal. , 75:51--58, 1980

1980
[40]

Kelleher and J

C. Kelleher and J. Streets. Entropy, stability, and yang-mills flow. Commun. Contemp. Math. , 18(2):1550032, 2016

2016
[41]

Kelleher and J

C. Kelleher and J. Streets. Singularity formation of the yang-mills flow. In Ann. Inst. Henri Poincaré C Anal. Non Linéaire , volume 35, pages 1655--1686, 2018

2018
[42]

H. Naito. Finite time blowing-up for the yang-mills gradient flow in higher dimensions. Hokkaido Math. J. , 23(3):451--464, 1994

1994
[43]

J. Rade. On the yang-mills heat equation in two and three dimensions. J. Reine Angew. Math. , 426:123--164, 1992

1992
[44]

Reed and B

M. Reed and B. Simon. Methods of modern mathematical physics; Analysis of operators . Marcourt Brace Jovanovich, 1978

1978
[45]

W. Schlag. Schauder and $l^p$ estimates for parabolic systems via campanato spaces. Comm. Partial Differential Equations , 21(7--8):1141--1175, 1996

1996
[46]

Schlatter

A. Schlatter. Global existence of the yang-mills flow in four dimensions. J. Reine Angew. Math. , 475:133--148, 1996

1996
[47]

Schlatter

A. Schlatter. Long-time behavior of the yang-mills flow in four dimensions. Ann. Global Anal. Geom. , 15(1):1--25, 1997

1997
[48]

N. Sesum. Limiting behavior of the ricci flow. arXiv preprint , math/0402194, 2004

2004 arXiv
[49]

N. Sesum. Convergence of the ricci flow toward a soliton. Comm. Anal. Geom. , 14(2):283--343, 2006

2006
[50]

N. Sesum. Linear and dynamical stability of ricci-flat metrics. Duke Math. J. , 131(1):1--26, 2006

2006
[51]

N. Sesum. Rate of convergence of the mean curvature flow. Comm. Pure Appl. Math. , 61(4):464--485, 2008

2008
[52]

L. Simon. Schauder estimates by scaling. Calc. Var. Partial Differential Equations , 5:391--407, 1997

1997
[53]

A. E. Schlatter, M. Struwe, and A. S. Tahvildar-Zadeh. Global existence of the equivariant yang-mills heat flow in four space dimensions. Amer. J. Math. , 120(1):117--128, 1998

1998
[54]

M. Struwe. The yang-mills flow in four dimensions. Calc. Var. Partial Differential Equations , 2:123--150, 1994

1994
[55]

Y. Sire, J. Wei, , and Y. Zheng. Infinite time bubbling for the su(2) yang--mills heat flow on R ^4 . arXiv:2208.13875 , 2022

2022 arXiv
[56]

K. K. Uhlenbeck. Removable singularities in yang-mills fields. Comm. Math. Phys. , 83:11--29, 1982

1982
[57]

A. Waldron. Self-duality and singularities in the yang-mills flow. Columbia University Thesis , 2014

2014
[58]

A. Waldron. Instantons and singularities in the yang-mills flow. Calc. Var. Partial Differential Equations , 55:1--31, 2016

2016
[59]

A. Waldron. Long-time existence for yang-mills flow. Invent. Math. , 217(3):1069--1147, 2019

2019
[60]

Weinkove

B. Weinkove. Singularity formation in the yang-mills flow. Calc. Var. Partial Differential Equations , 19(2):211--220, 2004

2004

Pith tools

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

[1] [1]

Angenent, S

S. Angenent, S. Brendle, P. Daskalopoulos, and N. Sesum. Unique asymptotics of compact ancient solutions to three-dimensional ricci flow. Comm. Pure Appl. Math. , 75(5):1032--1073, 2022

work page 2022

[2] [2]

Angenent, P

S. Angenent, P. Daskalopoulos, and N. Sesum. Unique asymptotics of ancient convex mean curvature flow solutions. J. Differential Geom. , 111(3):381--455, 2019

2019

[3] [3]

Angenent, P

S. Angenent, P. Daskalopoulos, and N. Sesum. Uniqueness of two-convex closed ancient solutions to the mean curvature flow. Ann. of Math. , 192(2):353--436, 2020

work page 2020

[4] [4]

S. B. Angenent and J. J. L. Velazquez. Degenerate neckpinches in mean curvature flow. J. Reine Angew. Math. , 482:15--66, 1997

work page 1997

[5] [5]

Brendle and K

S. Brendle and K. Choi. Uniqueness of convex ancient solutions to mean curvature flow in R ^3 . Invent. Math. , 217(1):35--76, 2019

work page 2019

[6] [6]

Brendle and K

S. Brendle and K. Choi. Uniqueness of convex ancient solutions to mean curvature flow in higher dimensions. Geom. Topol. , 25(5):2195--2234, 2021

2021

[7] [7]

Brendle, P

S. Brendle, P. Daskalopoulos, K. Naff, and N. Sesum. Uniqueness of compact ancient solutions to the higher-dimensional ricci flow. J. Reine Angew. Math. , 2023(795):85--138, 2023

work page 2023

[8] [8]

Brendle, P

S. Brendle, P. Daskalopoulos, and N. Sesum. Uniqueness of compact ancient solutions to three-dimensional ricci flow. Invent. Math. , 226(2):579--651, 2021

work page 2021