Improved explicit estimates for the discrete Laplace operator with hyperbolic circle patterns

Aijin Lin; Longxiang Wu

REVIEW 3 major objections 3 minor 33 references

Improved explicit estimates for the discrete Laplace operator with hyperbolic circle patterns

T0 review · 3 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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

Pith's one-line read For hyperbolic circle packings, the discrete Laplace operator's coefficients are bounded above by constants that depend only on the intersection-angle weights, not on the circle radii.

desk verdict The paper's removal of the radius lower bound for the discrete Laplacian estimate is a real result, but the proof of the boundary boundedness is not rigorous as written. read the letter →

arxiv 2507.03901 v1 pith:5LZWUXSV submitted 2025-07-05 math.DG

Aijin Lin , Longxiang Wu This is my paper · ORCID

classification math.DG MSC 52C2653C44

keywords combinatorialp-thCalabiflowsdiscreteLaplaceoperatorlongtimeexistencehyperboliccirclepatternsobtuseintersectionanglespackingmetricsuniformestimatescurvature

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

Geometric flows on circle-packing surfaces need control of a discrete Laplace operator whose coefficients $A_i$ and $B_{ij}$ depend on the hyperbolic circle radii. This paper proves two explicit estimates for those coefficients: a lower-radius version and, more strongly, a radius-free version. The main theorem states that when the edge weights $\Phi$ lie in $[0,\pi)$ and satisfy the obtuse-angle condition $(\star)$, the inequalities $0 < A_i \le C_1(\Phi)$ and $0 \le B_{ij} \le C_2(\Phi)$ hold for every circle packing metric. The interest is that the upper constants do not depend on the radii at all, so compactness along combinatorial Calabi flows in hyperbolic geometry can be obtained without an a priori lower bound on circle sizes. As applications, the paper gives new proofs of long-time existence for the combinatorial Calabi flow and the combinatorial $p$-th Calabi flow.

What carries the argument

The load-bearing identity is the duality relation of Lemma 2.1, expressed in hyperbolic triangle geometry: $\partial \theta_i^{jk}/\partial u_i = -\cosh \ell_{ij}\,\partial \theta_i^{jk}/\partial u_j - \cosh \ell_{ik}\,\partial \theta_i^{jk}/\partial u_k$, together with its corollary $A_i = \sum_{v_j \sim v_i} B_{ij}(\cosh \ell_{ij} - 1)$, which writes diagonal coefficients as sums of off-diagonal ones. The explicit estimates then come from a rational-function reduction: after substituting radius variables $x, y, z$, the quantities $T_1$ and $T_2$ take the form $f/(g_1\sqrt{g_2})$ with explicit polynomials $f, g_1, g_2$ whose nonnegative coefficients are controlled by condition $(\star)$. The uniform bound follows by analyzing each boundary regime of the positive octant.

What would settle it

Compute $T_1(x,y,z)$ from equation (4.12) along a sequence with $x = y^2$, $z$ fixed, and weights satisfying $(\star)$, letting $x \to 0$; if the values are unbounded, the claimed neighborhood boundedness fails. Alternatively, numerically construct a circle packing triangle with one radius tending to zero and a fixed weight $\Phi$, and check whether $A_i$ exceeds every constant depending only on $\Phi$.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 1.2: on a closed triangulated surface with weight function $\Phi: E \to [0,\pi)$ satisfying $(\star)$, every circle packing metric $r$ gives coefficients $A_i > 0$ and $B_{ij} \ge 0$ with uniform upper bounds $A_i \le C_1(\Phi)$ and $B_{ij} \le C_2(\Phi)$ depending only on $\Phi$ and the triangulation. The proof rewrites the angle derivatives $T_1 = \partial \theta_i^{jk}/\partial u_j$ and $T_2 = T_1(\cosh \ell_{ij} - 1)$, after the change $x = (e^{2r_i}-1)/2$, as rational functions on $(0,\infty)^3$ and establishes their uniform boundedness by splitting the domain into regions where zero, one, two, or three of the variables are small. This extends the known zero-weight estimate to all admissible obtuse weights and removes the lower-radius assumption from earlier compactness arguments.

Load-bearing premise

One load-bearing premise is that the rational functions encoding the Laplacian coefficients are bounded on the whole positive octant; the proof establishes this by checking limits along straight-line paths and then concludes the bound holds in a full neighborhood of the boundary.

Editorial extensions

If this is right

The discrete Laplace coefficients stay bounded even as some circle radii go to zero, so flow compactness does not require an artificial positive lower radius bound.
The combinatorial Calabi flow and the combinatorial $p$-th Calabi flow exist for all time in hyperbolic background geometry from any initial circle packing metric.
The special case $\Phi \equiv 0$ is recovered, so the radius-free estimate subsumes the earlier zero-weight estimate.
The bounds are expressed by explicit constants involving the weight function and the maximal vertex degree, making the compactness quantitative.

Reading between the lines

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

The same rational-function reduction could be adapted to Euclidean or inversive-distance circle patterns, where the Laplacian coefficients satisfy analogous identities, potentially yielding radius-free bounds there as well.
For triangulations with bounded degree, the constants $C_1(\Phi)$ and $C_2(\Phi)$ become uniform across entire families of surfaces, which may be useful in numerical circle-packing applications.
A more robust proof of the boundary boundedness of $T_1$ and $T_2$, for instance by a compactness argument on the sphere of directions, would strengthen confidence in Theorem 1.2 without changing its statement.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

Colleague,

You should know two things about this paper. The main new result, Theorem 1.2, is a uniform upper bound for the discrete Laplacian in hyperbolic circle patterns that depends only on the weight and triangulation, not on the circle radii. That is a real extension of Ge-Xu's zero-weight bound. The second thing is that the proof has a genuine gap, and there is also an algebraic error. I would not trust Theorem 1.2 as written.

What is solid: the paper honestly credits Xu-Wu for Theorem 1.1, and the new proof of it is explicit and direct. The application to long-time existence of combinatorial Calabi flows is standard once the estimates are in hand. The polynomial rescaling in Section 4 is a reasonable tactic, and the case breakdown is thorough in spirit.

The first problem is in equation (4.9). The expansion of ((x+1)(y+1)+φxy)^2 - (2x+1)(2y+1) gives (1+φ)^2 x^2y^2, not (3+4φ+φ^2)x^2y^2. This polynomial is used throughout Section 4. The fix is easy, and because the true coefficient is smaller than the one printed, the bounds would only become easier, but the printed algebra is wrong.

The bigger problem is in Case 1 of Section 4. The authors need to show that the rational function s on the unit quarter-sphere is bounded near (0,0,1). They check limits along rays ŷ = kx̂ for fixed k, note the limits as k→0 and k→∞, and then assert a uniform bound in a full neighborhood. That implication is false for a non-homogeneous rational function: the ratio x̂/ŷ can vary arbitrarily along nonlinear paths, and no compactness or homogeneity argument is supplied. The subcase analysis also misses the boundary point (0,1,0) when a1=0, where the denominator also vanishes. Since the final constants rest on this step, Theorem 1.2 is unsupported as written.

Minor point: Theorem 1.2 states 0 < B_ij, but the proof only yields B_ij ≥ 0, and equality can occur (e.g., Φij=0 with Φik=Φjk=π/2).

The citation pattern is fine; the paper is candid about sources. This is a paper for people working on combinatorial curvature flows. It deserves peer review, but the referee should demand a rigorous boundary argument and the algebra fix before acceptance.

My verdict: accept for review, but only as conditional.

Referee Report

3 major / 3 minor

Summary. The paper studies the discrete Laplace operator associated with hyperbolic circle patterns on closed triangulated surfaces with edge weights Φ∈[0,π) satisfying Zhou's condition (⋆). The authors give explicit bounds for the quantities Ai and Bij: Theorem 1.1 assumes a uniform lower radius ri≥R, and Theorem 1.2 claims uniform upper bounds independent of R, together with strict positivity of Ai and Bij. The proofs rewrite the relevant angle derivatives in exponential variables x,y,z, partition (0,∞)^3 into Ω1∪Ω2, and perform a case analysis near the boundary. Section 5 then uses these estimates to prove long-time existence of combinatorial Calabi and p-th Calabi flows. The appendix supplies an analytic verification of the Glickenstein-Thomas identity used in Section 2.

Significance. If the estimates were correct, they would be a valuable contribution: they extend Ge-Xu's uniform control for Φ≡0 to all weights satisfying (⋆), provide explicit constants depending only on the weight and the combinatorics, and give a new proof of long-time existence for combinatorial Calabi and p-th Calabi flows. The paper also contains useful explicit computations, including an analytic proof of the Glickenstein-Thomas relation. However, the main new theorem is not established as written: there is a concrete algebraic error in the polynomial g1, the localization argument near the origin in Case 1 is logically invalid, and the claimed strict positivity of Bij is false as stated. These issues are load-bearing but appear repairable, so the appropriate decision is major revision rather than rejection.

major comments (3)

[§4, Eq. (4.9)] Equation (4.9) contains a false algebraic identity. Direct expansion gives ((x+1)(y+1)+φijxy)^2 - (2x+1)(2y+1) = (1+φij)^2 x^2y^2 + 2(1+φij)(x^2y+xy^2) + x^2 + y^2 + 2φijxy, not (3+4φij+φij^2)x^2y^2 + ... . The incorrect coefficient propagates to (4.18), (4.30), (4.36) and (4.49), and it enters the definitions of T1 and T2 in (4.12)-(4.14). Because g1 is a denominator factor in all of Section 4, every estimate built on this expansion must be rechecked. The corrected polynomial is still positive, so the error may be fixable, but as written the central computation of Section 4 is wrong.
[§4, Case 1, between (4.20) and (4.22)] The proof that s(hat x, hat y, hat z) is bounded near (0,0,1) and (1,0,0) is invalid. The argument fixes k and studies the straight path hat y = k hat x; a bound along each such ray does not imply a uniform bound in a full neighborhood, because s is not controlled along nonlinear paths or as the ratio hat x/hat y varies. The assertion that 'along any path' the bound holds, and the passage to a ball B((0,0,1),δ0), require a compactness or dominated-convergence argument that is not supplied. In addition, in the subcase a3=0, a2≠0, the denominator of s also vanishes at (0,1,0), which is not treated; the text near (1,0,0) also refers to B((0,0,1),δ1), apparently a typo for B((1,0,0),δ1). Since (4.22) is the basis for the final constants M1 and M2, the uniform upper bound in Theorem 1.2 is not established as written.
[§4, Theorem 4.1 and Theorem 1.2] The strict positivity 0 < Bij is not supported by the proof and is in fact false in general. The proof only establishes Bij ≥ 0. A concrete example satisfying (⋆) is Φij = 0 and Φik = Φjk = Φil = Φjl = π/2 for the two triangles sharing edge ij; then the numerator in (3.2)/(4.2) is identically zero for both faces, so Bij = 0. This contradicts the statements of Theorem 1.2 and Theorem 4.1. The applications in Section 5 use only Bij ≥ 0, so the statements could be repaired by replacing '0 < Bij' with '0 ≤ Bij' (or by adding hypotheses), but the theorem as advertised is false.

minor comments (3)

[§4, Case 3, third bullet] The bullet 'z → 0, x → x̄ ≥ δ̄, z → ȳ ≥ δ̄' appears to contain a typo; the last condition should be y → ȳ ≥ δ̄.
[Theorem 1.2 vs. Theorem 4.1] Theorem 1.2 in the introduction says the constants depend only on Φ and the triangulation, while Theorem 4.1 says 'depending only on Φ'; the final constants in the proof use the maximal degree d, so the wording should be made consistent.
[Appendix, §6.1] In the first displayed notation line, Cij = cosh lij and Sij = cosh lij are both defined with the same symbol for the hyperbolic sine; Sij should be sinh lij.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Laplace estimates are derived from explicit hyperbolic trigonometric formulas, not from the long-time existence results they reprove.

full rationale

The derivation chain is self-contained in the relevant sense. Lemma 2.1 is attributed to Glickenstein–Thomas but is proved analytically in the appendix from the hyperbolic cosine law and the derivative symmetry of Chow–Luo, so the paper does not import its key structural identity as an unverified premise. The bounds on Ai and Bij in Theorems 3.2 and 4.1 are obtained by bounding explicit rational expressions (4.2)–(4.13) in terms of the variables x,y,z and weight-dependent polynomials; the constants are explicit functions of the weight and triangulation, not fitted parameters. The long-time existence result in Theorem 1.3 uses these new uniform estimates together with the external angle-shrinking lemma of Zhang–Zheng; it does not invoke the Ge–Xu, Ge–Hua, or Lin–Zhang existence theorems as premises. Self-citations to Lin–Zhang and related prior work are contextual or applications-oriented and are not load-bearing. There is a genuine mathematical gap in Section 4 where boundedness along straight-line paths is asserted to imply boundedness in a neighborhood without a valid uniformity argument, and the claimed strict positivity 0<Bij is not supported in cases where the stated equality conditions are attainable; these are correctness concerns, not circularity. No step in the derivation reduces, by construction or self-citation, to the target result.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central claim rests on structural assumptions about hyperbolic circle patterns (the (⋆) condition and known symmetry/angle-derivative identities). No new entities or fitted physical constants are introduced. The constants C1, C2, a1, a2, a3 are constructed explicitly from the weights and triangulation; the only hand-chosen auxiliary parameter is the cutoff δ.

free parameters (1)

delta (cutoff for domain partition) = sufficiently small, chosen depending on weights
In Section 4, equation (4.15), the domain (0,∞)^3 is partitioned into sets where min(x,y,z) ≥ δ or < δ. The constant δ is chosen 'sufficiently small' by hand, depending on the weights, to make the case analysis work. It is not fitted to data.

assumptions (4)

domain assumption Condition (⋆): for every triangle, cos Φij + cos Φik cos Φjk ≥ 0 cyclically.
Assumed throughout Section 1.1. Needed for triangle inequalities and for nonnegativity of angle derivative terms in the proofs of Theorems 1.1 and 1.2.
domain assumption Chow-Luo Lemma A1: the Jacobian ∂K_i/∂u_j is symmetric for weights in [0, π).
Invoked in Section 2 to justify the symmetry of the discrete Laplacian and the expressions for Ai and Bij. Established in earlier literature.
domain assumption Lemma 2.1 (Glickenstein-Thomas, Proposition 9) on angle derivative identities for hyperbolic circle patterns.
Used in Corollary 2.2 and throughout the estimates. An analytic proof is provided in the appendix, but the lemma is cited as a known structural result from [19].
domain assumption Lemma 5.2 (Zhang-Zheng [32]): for fixed weights satisfying (⋆), the inner angle θ_i^{jk} tends to 0 as the radius r_i tends to infinity.
Used in Section 5 to bound the p-th Calabi flow from above when radii become large.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Improved explicit estimates for the discrete Laplace operator with hyperbolic circle patterns." pith.science (2026). https://pith.science/paper/5LZWUXSV

@misc{pith2026250703901,
  author       = {Pith},
  title        = {Pith review of: Improved explicit estimates for the discrete Laplace operator with hyperbolic circle patterns},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5LZWUXSV}},
  note         = {Machine review of arXiv:2507.03901}
}

abstract

Ge in his thesis \cite{Ge-thesis} introduced the combinatorial Calabi flows and established the long time existence and convergence of solutions to the flows in both hyperbolic and Euclidean background geometries. It is noteworthy that the existence of solutions to the combinatorial Calabi flows in hyperbolic background geometry proves to be more intricate and challenging compared to the Euclidean background geometry. The main difficulty is to establish the compactness, especially the lower boundeness along the flow equations. In this paper, we give two explicit estimates for the discrete Laplace operator based on the Glickenstein-Thomas formulation \cite{Glickenstein2017} for discrete hyperbolic conformal structures. As applications, we give new proofs of the long time existence of solutions to the combinatorial Calabi flows established by Ge-Xu \cite{Ge2016}, Ge-Hua \cite{Ge2018} and the combinatorial $p$-th Calabi flows established by Lin-Zhang \cite{Lin2019} in hyperbolic background geometry.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

33 extracted references · 31 canonical work pages

[1]

Angle, T

O. Angle, T. Hutchcroft, A. Nachmias, G. Ray, Unimodular hyperbolic triangulation: circle packing and random walk , Invent. Math. 206 (2016) 229-268

work page 2016
[2]

Beardon, K

A. Beardon, K. Stephenson, Circle packings in different geometries , Tohoku Math. J. 43 (1991) 27-36

work page 1991
[3]

Calabi, Extremal K¨ ahler metrics, Ann

E. Calabi, Extremal K¨ ahler metrics, Ann. of Math. Stud., vol. 102, Princeton Univ. Press, Princeton, N.J., 1982, pp. 259-290. 27

work page 1982
[4]

Calabi, Extremal K¨ ahler metrics II, Differential geometry and complex analysis, Springer, Berlin, (1985) 95-114

E. Calabi, Extremal K¨ ahler metrics II, Differential geometry and complex analysis, Springer, Berlin, (1985) 95-114

work page 1985
[5]

Chang, Global existence and convergence of solutions of Calabi flow on surfaces of genus h ≥ 2 , Kyoto J

S. Chang, Global existence and convergence of solutions of Calabi flow on surfaces of genus h ≥ 2 , Kyoto J. Math. 40 (2000) 363-377

work page 2000
[6]

Chang, The 2-dimensional Calabi flow , Nagoya Math

S. Chang, The 2-dimensional Calabi flow , Nagoya Math. J. 181 (2006) 63-73

work page 2006
[7]

Chen, Calabi flow in Riemann surfaces revisited: a new point of view , Int

X. Chen, Calabi flow in Riemann surfaces revisited: a new point of view , Int. Math. Res. Not. (2001) 275-297

work page 2001
[8]

B. Chow, F. Luo, Combinatorial Ricci flows on surfaces , J. Differ. Geom. 63 (2003) 97-129

work page 2003

Show all 33 references

[9]

K. Feng, H. Ge, B. Hua, Combinatorial Ricci flows and the hyperbolization of a class of compact 3- manifolds, Geom. Topol. 26 (2022) 1349-1384

2022
[10]

K. Feng, A. Lin, X. Zhang, Combinatorial p-th Calabi flows for discrete conformal factors on surfaces J. Geom. Anal. 30 (2020) 3979-3994

2020
[11]

Ge, Combinatorial methods and geometric equations, Thesis (Ph.D.)-Peking University, Beijing

H. Ge, Combinatorial methods and geometric equations, Thesis (Ph.D.)-Peking University, Beijing. (2012). 144 pp

2012
[12]

Ge, Combinatorial Calabi flows on surfaces , Trans

H. Ge, Combinatorial Calabi flows on surfaces , Trans. Am. Math. Soc. 370 (2017) 1377-1391

2017
[13]

H. Ge, B. Hua, On combinatorial calabi flow with hyperbolic circle patterns , Adv. Math. 333 (2018), 523- 538

2018
[14]

H. Ge, W. Jiang, On the deformation of inversive distance circle packings I , Trans. Am. Math. Soc. 372 (2019) 6231-6261

2019
[15]

H. Ge, W. Jiang, On the deformation of inversive distance circle packings II , J. Funct. Anal. 272 (2017) 3573-3595

2017
[16]

H. Ge, W. Jiang, On the deformation of inversive distance circle packings III , J. Funct. Anal. 272 (2017) 3573-3595

2017
[17]

H. Ge, A. Lin, The character of Thurston ’s circle packings, Sci. China. Math. 67 (2024) 1869-1862

2024
[18]

H. Ge, X. Xu, 2-dimensional combinatorial calabi flow in hyperbolic background geometry , Differ. Geom. Appl. 47 (2016), 86-98

2016
[19]

Glickenstein, J

D. Glickenstein, J. Thomas, Duality structures and discrete conformal variations of piecewise constant curvature surfaces, Adv. Math. 320 (2017), 250-278

2017
[20]

Guo, Combinatorial Yamabe flow on hyperbolic surfaces with boundary , Comm

R. Guo, Combinatorial Yamabe flow on hyperbolic surfaces with boundary , Comm. Contemp. Math. 13 (2011) 827-842

2011
[21]

Hamilton, Three-manifolds with positive Ricci curvature , J

R. Hamilton, Three-manifolds with positive Ricci curvature , J. Differ. Geom. 17 (1982) 255-306

1982
[22]

G. Hu, Z. Lei, Y. Qi, P. Zhou, Combinatorial p-th Calabi Flows for Total Geodesic Curvatures in Hyperbolic Background Geometry, J. Geom. Anal. 35 (2025) 1-18

2025
[23]

Z. He, O. Schramm, Hyperbolic and parabolic circle packings, Discrete Comput. Geom. 14 (1995) 123-149

1995
[24]

A. Lin, X. Zhang, Combinatorial p-th Calabi flows on surfaces , Adv. Math. 346 (2019) 1067-1090

2019
[25]

A. Lin, X. Zhang, Combinatorial p-th Ricci flows on surfaces , Nonlinear Anal. 211 (2021) 112417, 12 pp

2021
[26]

Luo, Combinatorial Yamabe flow on surfaces , Commun

F. Luo, Combinatorial Yamabe flow on surfaces , Commun. Contemp. Math. 6 (2004) 765-780

2004
[27]

Luo, A combinatorial curvature flow for compact 3-manifolds with boundary , Electron

F. Luo, A combinatorial curvature flow for compact 3-manifolds with boundary , Electron. Res. Announc. Am. Math. Soc. 11 (2005) 12-20

2005
[28]

Thurston, Geometry and Topology of 3-Manifolds , Princeton Lecture Notes, 1976

W. Thurston, Geometry and Topology of 3-Manifolds , Princeton Lecture Notes, 1976

1976
[29]

Colin de Verdi` ere, Un principe variationnel pour les empilements de cercles , Invent

Y. Colin de Verdi` ere, Un principe variationnel pour les empilements de cercles , Invent. Math. 104 (3) (1991) 655-669

1991
[30]

Xu, Rigidity of inversive distance circle packings revisited , Adv

X. Xu, Rigidity of inversive distance circle packings revisited , Adv. Math. 332 (2018), 476-509

2018
[31]

T. Wu, X. Xu, Fractional combinatorial Calabi flow on surfaces . arXiv:2107.14102

arXiv
[32]

Zhang, X

X. Zhang, X. Zheng, On the deformation of Thurston ’s circle packings with obtuse intersection angles , J. Geom. Anal. 264 (2024) 1-24

2024
[33]

Zhou, Circle patterns with obtuse exterior intersection angles

Z. Zhou, Circle patterns with obtuse exterior intersection angles . arXiv:1703.01768. 28

arXiv

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Angle, T

O. Angle, T. Hutchcroft, A. Nachmias, G. Ray, Unimodular hyperbolic triangulation: circle packing and random walk , Invent. Math. 206 (2016) 229-268

work page 2016

[2] [2]

Beardon, K

A. Beardon, K. Stephenson, Circle packings in different geometries , Tohoku Math. J. 43 (1991) 27-36

work page 1991

[3] [3]

Calabi, Extremal K¨ ahler metrics, Ann

E. Calabi, Extremal K¨ ahler metrics, Ann. of Math. Stud., vol. 102, Princeton Univ. Press, Princeton, N.J., 1982, pp. 259-290. 27

work page 1982

[4] [4]

Calabi, Extremal K¨ ahler metrics II, Differential geometry and complex analysis, Springer, Berlin, (1985) 95-114

E. Calabi, Extremal K¨ ahler metrics II, Differential geometry and complex analysis, Springer, Berlin, (1985) 95-114

work page 1985

[5] [5]

Chang, Global existence and convergence of solutions of Calabi flow on surfaces of genus h ≥ 2 , Kyoto J

S. Chang, Global existence and convergence of solutions of Calabi flow on surfaces of genus h ≥ 2 , Kyoto J. Math. 40 (2000) 363-377

work page 2000

[6] [6]

Chang, The 2-dimensional Calabi flow , Nagoya Math

S. Chang, The 2-dimensional Calabi flow , Nagoya Math. J. 181 (2006) 63-73

work page 2006

[7] [7]

Chen, Calabi flow in Riemann surfaces revisited: a new point of view , Int

X. Chen, Calabi flow in Riemann surfaces revisited: a new point of view , Int. Math. Res. Not. (2001) 275-297

work page 2001

[8] [8]

B. Chow, F. Luo, Combinatorial Ricci flows on surfaces , J. Differ. Geom. 63 (2003) 97-129

work page 2003