Pith. sign in

REVIEW 2 major objections 5 minor 35 references

Gradient estimates and parabolic frequency monotonicity for positive solutions of the heat equation under generalized Ricci flow

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read This paper proves Li-Yau-type and Hamilton-type gradient estimates for positive heat-equation solutions under the generalized Ricci flow, and derives Harnack inequalities and parabolic-frequency monotonicity from them.

desk verdict A real factor-3 algebra error in the main Li-Yau proof invalidates the stated constants, but the approach is sound and the paper deserves refereeing after correction. read the letter →

arxiv 2506.04937 v1 pith:E4MUXCJJ submitted 2025-06-05 math.DG

Juanling Lu , Yu Zheng This is my paper
classification math.DG MSC 53E2058J3535K05
keywords Li-YaugradientestimateHamilton-typegeneralizedRicciflowheatequationHarnackinequalityparabolicfrequencydifferentialconjugatekernel
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

This paper extends the classical Li-Yau and Hamilton gradient estimates from the Ricci flow to the generalized Ricci flow, in which the metric evolves by $-2\mathrm{Ric}+\tfrac12 H^2$ and the closed three-form $H$ evolves by its natural adjoint equation. The main theorem gives, for a positive solution of the heat equation, an upper bound on $|\nabla u|^2/u^2-\alpha\,\partial_t u/u$ under the curvature assumptions $-K_1 g/t\le\mathrm{Ric}\le K_2 g/t$ and $H^2\le K_3 g/t$, with $\alpha>1$. Because the curvature bounds are allowed to blow up like $1/t$ near the initial time, the hypothesis is weaker than the uniform curvature bounds used in earlier Ricci-flow estimates. If the theorem is correct, integrating the bound yields a spacetime Harnack inequality, and a separate Hamilton-type estimate gives the curvature-independent bound $|\nabla u|^2\le (u^2/t)\ln(A/u)$. The final application is a parabolic frequency $U(t)$ whose monotonicity along the flow controls the first weighted eigenvalue and yields an integral Harnack inequality.

What carries the argument

The load-bearing object is the functional $F=t(|\nabla f|^2-\alpha\partial_t f)$ for $f=\ln u$, whose evolution is computed in Lemma 2.1. The proof's algebraic engine is a square completion in the Hessian, using parameters $a,b>0$ with $a+2b=1/\alpha$ to rewrite the Ricci and $H^2$ terms as sums of squares plus remainders, followed by the identity $(y-z)^2=\alpha^{-2}(y-\alpha z)^2+((\alpha-1)/\alpha)^2y^2+2(\alpha-1)\alpha^{-2}y(y-\alpha z)$ that converts the $\Delta f$ term into the quantity $F$ plus controlled positive terms. Lemma 2.2, $|\nabla(\operatorname{tr}Q)|^2\le n|\nabla Q|^2$, controls the $H$-flux terms through the assumption $|\nabla H^2|\le K_4$. A maximum-principle argument at the maximum point of $F$ then yields a quadratic inequality in $F/t$, and solving it gives Theorem 1.1. For the frequency part, the machinery is the weighted measure $d\mu_{g(t)}=K\,dV$ with $K$ the conjugate heat kernel under the flow, and the frequency $U(t)=\exp\{E(t)\}h(t)\int|\nabla u|^2d\mu/\int u^2d\mu$, whose derivative is controlled by the gradient estimates.

What would settle it

Substitute the identity $(y-z)^2=\alpha^{-2}(y-\alpha z)^2+((\alpha-1)/\alpha)^2y^2+2(\alpha-1)\alpha^{-2}y(y-\alpha z)$ into (2.10) and compare the coefficient of $t|\nabla f|^2$ with $-3\alpha/8$; if the coefficient comes out as $-\alpha/8$ instead, the factor 3 is missing and the constants $B_1,B_3$ in Theorem 1.1 need rescaling.

Watch

Extended reading notes

Core claim

The paper's central claim is a quantitative differential Harnack inequality for positive solutions of $\partial_t u=\Delta_{g(t)}u$ under the generalized Ricci flow on a closed $n$-manifold. Writing $f=\ln u$ and $F=t(|\nabla f|^2-\alpha\partial_t f)$, Theorem 1.1 asserts that $F$ is bounded above in terms of $n,\alpha,a,b,K_1,K_2,K_3,K_4$, with $a+2b=1/\alpha$, whenever $-K_1 g/t\le\mathrm{Ric}\le K_2 g/t$ and $H^2\le K_3 g/t$. Equivalently, $|\nabla u|^2/u^2-\alpha\partial_t u/u\le n\alpha/(2a t)+\sqrt{n\alpha B_1/(2a)}+\sqrt{n\alpha B_2/(2a)}/t+\sqrt{n\alpha B_3/(2a t)}$ for explicit constants $B_1,B_2,B_3$. Corollary 1.5 integrates this along a geodesic to obtain a Harnack inequality comparing $u(x,t_1)$ and $u(y,t_2)$. The paper also proves a Hamilton-type bound $|\nabla u|^2\le (u^2/t)\ln(A/u)$ with $A=\max_M u(\cdot,0)$, and, in Theorem 1.9, shows that the parabolic frequency $U(t)=\exp\{E(t)\}D(t)/I(t)$ is monotone along the flow, with the direction determined by the sign of a time-dependent function $h(t)$. It claims these results extend and improve the corresponding Ricci-flow estimates because the Ricci curvature hypothesis is only a one-over-time bound rather than a uniform bound.

Load-bearing premise

Everything rests on the displayed algebraic step that rewrites equation (2.10) as (2.11); the coefficient shown there appears to cancel only a fraction of the term it should cancel, and if a factor of 3 is missing the constants $B_1$ and $B_3$ in the main bound and its corollaries must be revised.

Editorial extensions

If this is right

  • If Theorem 1.1 is correct, every positive heat-equation solution under the generalized Ricci flow with the stated curvature bounds satisfies an explicit differential Harnack inequality without a uniform-in-time Ricci bound.
  • Corollary 1.5 bounds the ratio $u(x,t_1)/u(y,t_2)$ by a geodesic-length exponential, making solution values at different spacetime points quantitatively comparable.
  • Theorem 1.7 gives the universal curvature-free gradient bound $|\nabla u|^2\le (u^2/t)\ln(A/u)$ for closed manifolds.
  • Theorem 1.9 yields monotonicity of the parabolic frequency in both directions according to the sign of $h(t)$, hence monotonicity of the weighted first eigenvalue and an integral Harnack inequality.
  • In the special case $H\equiv0$, the generalized Ricci flow reduces to the Ricci flow and the estimates specialise to improvements of earlier Ricci-flow gradient estimates, since the curvature condition is relaxed from a constant bound to a $1/t$ bound.

Reading between the lines

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

  • An unstated consequence is that the $K/t$ curvature assumption makes these estimates natural tools for type-I singularity analysis: one could test whether the $F$-bound stays finite along a blow-up sequence of the generalized Ricci flow.
  • The evolution identity for $F$ is structural and only uses the form of the flow equations, so analogous Li-Yau-type estimates should be derivable for other flows whose metric evolution contains a positive quadratic tensor term, such as pluriclosed flow or Laplacian-type flows.
  • The monotone parabolic frequency points toward backward uniqueness for the heat equation under the generalized Ricci flow, the standard next step already taken for Ricci flow but not stated in this paper.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper studies positive solutions of the heat equation coupled with the generalized Ricci flow on a closed manifold, under curvature bounds of the form -K1/t g <= Ric <= K2/t g and H^2 <= K3/t g. It claims a Li-Yau-type differential Harnack estimate (Theorem 1.1) with explicit constants B1, B2, B3, a Hamilton-type estimate (Theorem 1.7), a Harnack inequality (Corollary 1.5), and a parabolic frequency monotonicity theorem (Theorem 1.9) with a specially chosen exponential weight E(t). The proofs use the Bochner formula, a weighted squared-tensor trick, and a maximum principle argument.

Significance. If correct, the paper would extend well-known gradient estimates and frequency monotonicity from the Ricci flow to the generalized Ricci flow under weaker, time-singular curvature bounds. The Hamilton-type estimate (Theorem 1.7) is standard and appears correct. The paper is self-contained, supplies explicit constants, and positions its results as improvements over earlier work by Liu and by Li-Li-Xu. However, the central Li-Yau-type estimate contains an algebraic error that changes the stated constants, so the quantitative claims in Theorem 1.1, its corollaries, and the frequency theorem are not supported as written.

major comments (2)
  1. [Section 2, Eq. (2.11)] The equality in (2.11) is false. The bracket on the right contains the term -(n/(16a))|\nabla f|^2, which, when multiplied by the prefactor 2a\alpha\,t/n, yields -\alpha t/8\,|\nabla f|^2. The left-hand side of the same equation, taken from (2.10), contains -3\alpha t/8\,|\nabla f|^2. Hence the coefficient is off by a factor of 3; to match (2.10), the bracket would need -(3n/(16a))|\nabla f|^2. This error propagates into the completing-the-square step: B1 in (2.12) changes by a factor of 9 (the n\alpha^3/(512a(\alpha-1)^2) term becomes 9n\alpha^3/(512a(\alpha-1)^2)) and B3 changes by a factor of 3, while B2 is unchanged. Consequently, Theorem 1.1's explicit bound, Corollary 1.5's Harnack inequality, and the constants C1 and C3 used in (3.2)-(3.4) for Theorem 1.9 are not correct as stated. The proof strategy is salvageable with revised constants, but the displayed quantitative results must be corrected.
  2. [Remark 1.4 and Eq. (1.3)] The claim that (1.3) follows from Theorem 1.1 when H=0 is not justified. Even with K3=K4=0, Theorem 1.1's bound contains the positive terms sqrt(n\alpha B1/(2a)) and (1/sqrt(t)) sqrt(n\alpha B3/(2a)), with B1=n\alpha^3/(512a(\alpha-1)^2) and B3=n\alpha^3 K1/(16a(\alpha-1)^2). These terms are absent from (1.3), and they cannot be discarded in an upper bound. If the H=0 case is meant to be handled by redoing the proof without the H-dependent Cauchy estimate (2.9), that separate computation should be provided; otherwise the claimed improvement over [25] in Remark 1.4 and (1.4) is unsupported.
minor comments (5)
  1. [Section 2, proof of Theorem 1.1] In the display after the quadratic formula, the bound is written as F(x,t) <= n\alpha/(2a) + sqrt(2)/2 sqrt(n\alpha/a)(B1 sigma^2 + B2 + B3 sigma); a square root around the last parenthesis is missing, and it should read sqrt(B1 sigma^2 + B2 + B3 sigma).
  2. [Throughout] There are several typos and minor language issues, including "the the time-dependent metric" in the introduction, "Harnack-hype inequality" before the proof of Corollary 1.5, and "forementioned" in Remark 1.10.
  3. [Section 3, after Eq. (3.2)] The text states "the constants K1, K2, K3, K4, K5 are as shown in Theorem 1.1," but Theorem 1.1 involves only K1, K2, K3, K4; the symbol K5 is never defined.
  4. [Corollary 1.5] The displayed Harnack inequality has an unbalanced parenthesis in the exponential factor; the intended grouping should be checked and typeset consistently.
  5. [Reference [16]] The title listed for Kopfer and Streets appears to be incorrect; it is listed as "Geometric measure of singular sets of elliptic equations," which does not match the known paper by these authors in J. Funct. Anal.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: all constants are derived, the frequency functional uses a legitimate Lyapunov construction, and there is no load-bearing self-citation.

full rationale

The derivation chain is self-contained. Theorem 1.1 is obtained from Lemma 2.1, a direct Bochner-type computation for F = t(|∇f|^2 - α∂_t f), together with Lemma 2.2 and the stated curvature bounds; the constants B1, B2, B3 emerge from Cauchy-Schwarz, completing the square, and the maximum principle rather than being fitted to the claimed bound. The Harnack inequality Corollary 1.5 and the Hamilton estimate Theorem 1.7 are standard maximum-principle consequences with no fitted parameters. In Section 3, the exponential factor E(t) in the frequency U(t) = exp{E(t)} D(t)/I(t) is explicitly defined by equation (3.2) precisely so that E'(t)h(t) + h'(t) cancels the term h(t)C(t) appearing in the derivative estimate; this is a legitimate Lyapunov-style construction, not a definition of U that assumes monotonicity. Monotonicity then follows from the independently proved gradient estimates (3.4) and (3.5). There are no load-bearing self-citations: comparisons with [25] and [35] are external and are not used to justify the main claims. The manuscript even flags α ≤ 1 as open (Remark 1.2), which is consistent with a genuine derivation rather than a forced conclusion. The possible factor-3 algebra issue in (2.10)-(2.11) is a correctness concern, not a circularity, since it does not make the theorem's conclusion equivalent to an input; the overall argument structure remains a forward derivation from assumptions.

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

The main input assumptions are the curvature and H bounds, plus standard parabolic theory. The only questionable item is the algebraic reduction in the proof of Theorem 1.1, which is a paper-specific step and currently inconsistent.

assumptions (5)
  • domain assumption Generalized Ricci flow solution exists on [0,T] with -K1/t g ≤ Ric ≤ K2/t g, H^2 ≤ K3/t g, |∇H^2| ≤ K4.
    Explicit hypotheses of Theorem 1.1 and Theorem 1.9.
  • domain assumption u is a smooth positive solution of ∂_t u = ∆_{g(t)} u.
    The object of study; positivity is used for f = ln u.
  • standard math Maximum principle for the parabolic operator ∂_t - ∆ on closed manifolds with time-dependent metric.
    Used in proofs of Theorems 1.1 and 1.7.
  • standard math Bochner formula and the evolution equation for ∆f under generalized Ricci flow (Lemma 2.1).
    Background identities; Lemma 2.1 is stated and proved in the paper.
  • ad hoc to paper The algebraic identity in (2.11) relating (2.10) to the F-heavy expression.
    This step is specific to the paper and appears to contain an error: the factor n/(16a) should be 3n/(16a) to match (2.10).

how reviews work

0 comments
Cite this review

Pith. "Pith review of Gradient estimates and parabolic frequency monotonicity for positive solutions of the heat equation under generalized Ricci flow." pith.science (2026). https://pith.science/paper/E4MUXCJJ

@misc{pith2026250604937,
  author       = {Pith},
  title        = {Pith review of: Gradient estimates and parabolic frequency monotonicity for positive solutions of the heat equation under generalized Ricci flow},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E4MUXCJJ}},
  note         = {Machine review of arXiv:2506.04937}
}
read the original abstract

In this paper, we establish Li-Yau-type and Hamilton-type estimates for positive solutions to the heat equation associated with the generalized Ricci flow, under a less stringent curvature condition. Compared with [25] and [35], these estimates generalize the results in Ricci flow to this new flow under the weaker Ricci curvature bounded assumption. As an application, we derive the Harnack-type inequalities in spacetime and find the monotonicity of one parabolic frequency for positive solutions of the heat equation under bounded Ricci curvature.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

35 extracted references · 35 canonical work pages

  1. [16]

    Geometric measure of singular sets of elliptic equations

    Kopfer, E., and J. Streets. “ Geometric measure of singular sets of elliptic equations.”J. Funct. Anal.284,no. 10 (2023): 42, Paper No. 109901

    work page 2023

  2. [3]

    Gradient estimates for the heat equation under the Ricci flow

    B˘ aile¸ steanu, M., X. D. Cao, and A. Pulemotov. “Gradient estimates for the heat equation under the Ricci flow.”J. Funct. Anal.258, no. 10 (2010): 3517-3542

    work page 2010

  3. [25]

    Gradient estimates for solutions of the heat equation under Ricci flow

    Liu, S. P. “Gradient estimates for solutions of the heat equation under Ricci flow.”Pacific J. Math.243, no. 1 (2009): 165-180

    work page 2009

  4. [35]

    Some gradient estimates for the heat equation on domains and for an equation by Perelman

    Zhang, Q. S. “Some gradient estimates for the heat equation on domains and for an equation by Perelman.”Int. Math. Res. Not.(2006): 39, Art. ID 92314. 17

    work page 2006

  5. [17]

    Parabolic frequency monotonicity on Ricci flow and Ricci- harmonic flow with bounded curvatures

    Li, C. H., Y. Li, and K. R. Xu. “Parabolic frequency monotonicity on Ricci flow and Ricci- harmonic flow with bounded curvatures.”J. Geom. Anal.33, no. 9 (2023): 21, Paper No. 282

    work page 2023

  6. [1]

    Dirichlet’s problem for multiple valued functions and the regularity of mass minimizing integral currents

    Almgren, F. J. “Dirichlet’s problem for multiple valued functions and the regularity of mass minimizing integral currents.” InMinimal Submanifolds and Geodesics, 1-6. Amsterdam- New York: North-Holland, 1979. 15

    work page 1979

  7. [2]

    Gradient estimates for the heat equation under the Ricci-harmonic map flow

    B˘ aile¸ steanu, M. “Gradient estimates for the heat equation under the Ricci-harmonic map flow.”Adv. Geom.15, no. 4 (2015): 445-454

    work page 2015

  8. [4]

    Parabolic frequency on Ricci flows

    Baldauf, J., and D. Kim. “Parabolic frequency on Ricci flows.”Int. Math. Res. Not. IMRN no. 12 (2023): 10098-10114

    work page 2023

Show all 35 references
  1. [5]

    Parabolic frequency for the mean curvature flow

    Baldauf, J., P. T. Ho, and T.-K. Lee. “Parabolic frequency for the mean curvature flow.” Int. Math. Res. Not. IMRNno. 10 (2024): 8122-8136

    2024

  2. [6]

    Lu, and L

    Chow, B., P. Lu, and L. Ni.Hamilton ’s Ricci flow. Grad. Stud. Math., 77 American Math- ematical Society, Providence, RI; Science Press Beijing, New York, 2006

    2006

  3. [7]

    Harmonic functions with polynomial growth

    Colding, T. H., and W. P. Minicozzi. “Harmonic functions with polynomial growth.”J. Differential Geom.46, no. 1 (1997): 1-77

    1997

  4. [8]

    Parabolic frequency on manifolds

    Colding, T. H., and W. P. Minicozzi. “Parabolic frequency on manifolds.”Int. Math. Res. Not. IMRNno. 15 (2022): 11878-11890

    2022

  5. [9]

    Ricci flow, Killing spinors, and T-duality in generalized geometry

    Garcia-Fernandez, M. “Ricci flow, Killing spinors, and T-duality in generalized geometry.” Adv. Math.350, (2019): 1059-1108

    2019

  6. [10]

    Streets.Generalized Ricci flow

    Garcia-Fernandez, M., and J. Streets.Generalized Ricci flow. Univ. Lecture Ser., 76 Amer- ican Mathematical Society, Providence, RI, 2021

    2021

  7. [11]

    Monotonicity properties of variational integrals,A p weights and unique continuation

    Garofalo, N., and F.-H. Lin. “Monotonicity properties of variational integrals,A p weights and unique continuation.”Indiana Univ. Math. J.35, no. 2 (1986): 245-268

    1986

  8. [12]

    Unique continuation for elliptic operators: a geometric- variational approach

    Garofalo, N., and F.-H. Lin. “Unique continuation for elliptic operators: a geometric- variational approach.”Comm. Pure Appl. Math.40, no. 3 (1987): 347-366

    1987

  9. [13]

    A matrix Harnack estimate for the heat equation

    Hamilton, R. S. “A matrix Harnack estimate for the heat equation.”Comm. Anal. Geom. 1, no. 1 (1993): 113-126

    1993

  10. [14]

    Geometric measure of singular sets of elliptic equations

    Han, Q., R. Hardt, and F.-H. Lin. “ Geometric measure of singular sets of elliptic equations.” Comm. Pure Appl. Math.51, no. 11-12 (1998): 1425-1443

    1998

  11. [15]

    On the shrinking entropy functional for generalized Ricci flow

    Ishida, M. “ On the shrinking entropy functional for generalized Ricci flow.”J. Geom. Anal. 35, no. 5 (2025): 45, Paper No. 148

    2025

  12. [18]

    Gradient estimates and parabolic frequency under the LaplacianG 2 flow

    Li, C. H., Y. Li, and K. R. Xu. “Gradient estimates and parabolic frequency under the LaplacianG 2 flow.”Calc. Var. Partial Differential Equations64, no. 4 (2025): 28, Paper No. 121

    2025

  13. [19]

    On the parabolic kernel of the Schr¨ odinger operator

    Li, P., and S.-T. Yau. “On the parabolic kernel of the Schr¨ odinger operator.”Acta Math. 156, no. 3-4 (1986): 153-201. 16

    1986

  14. [20]

    Parabolic frequency monotonicity on compact manifolds

    Li, X. L., and K. Wang. “ Parabolic frequency monotonicity on compact manifolds.”Calc. Var. Partial Differential Equations58, no. 6 (2019): 18, Paper No. 189

    2019

  15. [21]

    Matrix Li-Yau-Hamilton estimates under Ricci flow and parabolic frequency

    Li, X. L., and Q. S. Zhang. “ Matrix Li-Yau-Hamilton estimates under Ricci flow and parabolic frequency.”Calc. Var. Partial Differential Equations63, no. 3 (2024): 38, Paper No. 63

    2024

  16. [22]

    Entropy and heat kernel on generalized Ricci flow

    Li, X. L. “ Entropy and heat kernel on generalized Ricci flow.”J. Geom. Anal.34, no. 2 (2024): 23, Paper No. 42

    2024

  17. [23]

    Nodal sets of solutions of elliptic and parabolic equations

    Lin, F.-H. “ Nodal sets of solutions of elliptic and parabolic equations.”Comm. Pure Appl. Math.44, no. 3 (1991): 287-308

    1991

  18. [24]

    A note on parabolic frequency and a theorem of Hardy-P´ olya-Szeg¨ o

    Liu, H.-Y., and P. Xu. “A note on parabolic frequency and a theorem of Hardy-P´ olya-Szeg¨ o.” Internat. J. Math.33, no. 9 (2022): 15, Paper No. 2250064

    2022

  19. [26]

    A Harnack inequality for parabolic differential equations

    Moser, J. “A Harnack inequality for parabolic differential equations.”Comm. Pure Appl. Math.17, (1964): 101-134

    1964

  20. [27]

    In: Analysis, complexgeometry, and mathematical physics: in honor of Duong H

    Ni, L.Parabolic frequency monotonicity and a theorem of Hardy-P´ olya-Szeg¨ o. In: Analysis, complexgeometry, and mathematical physics: in honor of Duong H. PhongContemp. Math., American Mathematical Society, Providence, RI, 644, (2015): 203-210

    2015

  21. [28]

    A gradient flow for worldsheet nonlinear sigma models

    Oliynyk, T., V. Suneeta, and E. Woolgar. “A gradient flow for worldsheet nonlinear sigma models.”Nuclear Phys.B 739, no. 3 (2006): 441-458

    2006

  22. [29]

    Unique continuation for parabolic equations

    Poon, C.-C. “Unique continuation for parabolic equations.”Comm. Partial Differential Equations21, no. 3-4 (1996): 521-539

    1996

  23. [30]

    Generalized geometry, T-duality, and renormalization group flow

    Streets, J. “Generalized geometry, T-duality, and renormalization group flow.”J. Geom. Phys.114, (2017): 506-522

    2017

  24. [31]

    Scalar curvature, entropy, and generalized Ricci flow

    Streets, J. “Scalar curvature, entropy, and generalized Ricci flow.”Int. Math. Res. Not. IMRNno. 11 (2023): 9481-9510

    2023

  25. [32]

    A parabolic flow of pluriclosed metrics

    Streets, J., and G. Tian. “ A parabolic flow of pluriclosed metrics.”Int. Math. Res. Not. IMRNno. 16 (2010): 3101-3133

    2010

  26. [33]

    Generalized K¨ ahler geometry and the pluriclosed flow

    Streets, J., and G. Tian. “ Generalized K¨ ahler geometry and the pluriclosed flow.”Nuclear Phys.B 858, no. 2 (2012): 366-376

    2012

  27. [34]

    Regularity results for pluriclosed flow

    Streets, J., and G. Tian. “ Regularity results for pluriclosed flow.”Geom. Topol.17, no. 4 (2013): 2389-2429

    2013

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.