Pith. sign in

REVIEW 3 major objections 5 minor 1 cited by

A prescribed curvature flow on hyperbolic surfaces with infinite topological type

T0 review · 3 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 An infinite prescribed curvature flow converges to generalized circle packing metrics with prescribed total geodesic curvatures on noncompact hyperbolic surfaces.

desk verdict A promising flow construction for infinite hyperbolic surfaces with a serious gap in the uniqueness proof; the paper deserves refereeing but needs a real revision. read the letter →

arxiv 2505.20091 v1 pith:EP72I5QY submitted 2025-05-26 math.GT math.DG

Xinrong Zhao , Puchun Zhou This is my paper
classification math.GTmath.DG MSC 52C2653E20
keywords prescribedcurvatureflowgeneralizedcirclepackingtotalgeodesichyperbolicsurfacesnoncompactinfinitecellulardecompositionboundariescusps
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 introduces an infinite prescribed curvature flow, a discrete analogue of Ricci flow tailored to infinite polygonal cellular decompositions of noncompact surfaces. It claims that this flow has global solutions for arbitrary initial data and is unique whenever the total geodesic curvatures stay uniformly bounded and face degree is bounded. Under two side conditions on the initial data and the prescribed curvatures, the flow converges to a generalized circle packing metric realizing exactly those total geodesic curvatures. In the triangulated case this gives smooth hyperbolic surfaces of infinite topological type with geodesic boundaries or cusps. The intended payoff is a flow-based construction of hyperbolic metrics on noncompact surfaces that has no other known construction.

What carries the argument

The flow itself is the central object: $s_i(t)=\ln k_i(t)$ evolves by the negative of the discrepancy $T_i-\hat T_i$, with $T_i$ assembled from per-face arcs $T_{i,P}$ given by formula (2.1). The proof machinery is an exhaustion argument: restrict the flow to finite subcomplexes, obtain uniform $C^2$ estimates from the maximum principle and curvature bounds, then pass to a diagonal limit. Uniqueness is obtained by writing the difference of two solutions as a weighted graph Laplacian plus a negative zero-order term and applying a maximum principle for infinite graphs, with the key estimate being a uniform bound on the Laplacian weights.

What would settle it

Evaluate the left-hand side of Lemma 3.13 numerically from formula (2.1) for a triangle with $k_P=2$ and $k_i$ ranging up to $10^3$; if $|\sum_{j\sim i}\partial T_i/\partial s_j|/T_i$ is unbounded as $k_i\to\infty$, the uniform bound fails and the uniqueness theorem for infinite flows is not established.

Watch

Extended reading notes

Core claim

The central claim is that the system $ds_i/dt = -(T_i - \hat T_i)$ on an infinite vertex set $V$, with $s_i=\ln k_i$ and $T_i$ the total geodesic curvature of the generalized circle at vertex $i$, is well-posed: existence for every $s_0$, and uniqueness among solutions with uniformly bounded total curvatures when face degree is bounded. The paper's two convergence theorems show that, if the initial discrepancy $T_i(s_0)-\hat T_i$ is everywhere nonnegative (Theorem 1.4) or everywhere nonpositive with the prescribed vector satisfying the finite-subset inequality (1.3) (Theorem 1.7), the flow converges to a generalized circle packing metric whose total geodesic curvatures are exactly $\hat T$. The corollary for infinite triangulations is that every prescribed curvature vector with $\hat T_v\le \deg(v)$ is realized, and because the flow keeps $s_i\le 0$ the realizing circles are horocycles or hypercycles, so the glued surface is smooth and noncompact with geodesic boundaries or cusps.

Load-bearing premise

The uniqueness half depends on the uniform estimate $|\sum_{j\sim i}\partial T_i/\partial s_j|\le C T_i$ proved in Lemma 3.13; the proof of that lemma appears to use formula (2.1) with the factor $k_i\sqrt{k_i^2-1}$ multiplied rather than divided, and if the estimate does not hold the uniqueness statement is unsupported.

Editorial extensions

If this is right

  • Global solutions of the prescribed curvature flow exist on every infinite polygonal cellular decomposition, so the flow is a well-defined deformation of generalized circle packing metrics on noncompact surfaces.
  • With bounded face degree and uniformly bounded total curvatures, the solution is unique, matching the uniqueness phenomenon known for continuous Ricci flows on noncompact manifolds.
  • When initial total curvature is at least the prescribed value everywhere, convergence to the prescribed packing is guaranteed; this is the route to Corollary 1.5.
  • For infinite triangulations, any prescribed vertex curvatures not exceeding vertex degree are realized by packings of horocycles and hypercycles, producing smooth hyperbolic surfaces of infinite topological type with infinitely many geodesic boundaries or cusps.
  • The second convergence theorem supplies existence under the finite-subset sum condition (1.3), and the simple-decomposition version weakens the condition to (1.4).

Reading between the lines

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

  • [editorial] The numerical content of (1.3) suggests a sharpness question: whether the $\pi\min\{N(P,W),N(P)-2\}$ budget for each face is also necessary for convergence in the nonpositive case; the paper does not address necessity.
  • [editorial] A direct check of Lemma 3.13 using formula (2.1) is the first place to test the uniqueness claim; if the uniform weight bound fails for large $k_i$, the existence and convergence theorems could survive while uniqueness needs a different argument.
  • [editorial] The construction yields surfaces whose conformal boundary data are encoded in the prescribed curvatures; a natural extension would be to use the same flow to realize prescribed curvatures in Euclidean or spherical background geometry, following the finite-case analogues.
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

3 major / 5 minor

Summary. The paper introduces an infinite prescribed curvature flow for generalized circle packing metrics on infinite polygonal cellular decompositions in hyperbolic background geometry. It claims global well-posedness of the flow, a uniqueness theorem under bounded total geodesic curvatures, and two convergence theorems that produce generalized circle packings with prescribed total geodesic curvatures, including surfaces of infinite topological type with geodesic boundaries or cusps. The proofs use exhaustion by finite subcomplexes, maximum principles, and imported results from prior work of Hu-Qi-Sun-Zhou and Ge-Hua-Zhou.

Significance. If the main results hold, the paper would meaningfully extend the combinatorial Ricci flow / prescribed curvature flow framework from finite triangulations to infinite cellular decompositions, providing a parabolic construction of noncompact hyperbolic surfaces of infinite type. The use of exhaustion and maximum-principle arguments is natural, and the connection to prescribed total geodesic curvature is a timely topic. The existence and convergence parts are presented as plausible consequences of the finite theory, but the uniqueness half of the well-posedness claim rests on a specific estimate in Lemma 3.13 that is not proven as written; this undermines the full well-posedness statement, although the existence and convergence results may be salvageable with a repaired argument.

major comments (3)
  1. [§3.2, Lemma 3.13] The proof of Lemma 3.13 is internally inconsistent with the definition (2.1). In case (1), for k_i>1, the proof writes T_i = 2 k_i sqrt(k_i^2-1) arccot(k_P sqrt(k_i^2-1)), while (2.1) gives T_{i,P} = 2 arccot(k_P sqrt(k_i^2-1))/(k_i sqrt(k_i^2-1)). Case (3) similarly omits the denominator factor 2/(k_i sqrt(1-k_i^2)). This is not a harmless typo: with the correct formula, for a one-parameter family with fixed k_P>1 and k_i→∞, the quantity being bounded behaves like 2(k_P^2-1)/(k_P k_i), while T_{i,P} behaves like 2/(k_P k_i^3), so the asserted uniform bound |Σ_{j∼i} ∂T_i/∂s_j| ≤ C T_i cannot hold for the printed expression. Since Lemma 3.13 is the only source of the uniform weight bound (3.7) used in the maximum-principle argument for Theorem 3.9, the uniqueness part of Theorems 1.2 and 3.9 is not substantiated as written.
  2. [§1, Corollary 1.3] Corollary 1.3 claims uniqueness for arbitrary initial values s0 under bounded vertex and face degree, but Theorem 3.9 requires the total geodesic curvatures of the two solutions to be uniformly bounded on V×[0,M]. If the prescribed vector T̂ is unbounded, the maximum principle in Proposition 3.8 only bounds T_i(t)-T̂_i, not T_i(t) itself, so the hypotheses of Theorem 3.9 need not hold. The corollary therefore does not follow from the stated uniqueness theorem unless an additional condition such as boundedness of T̂ (or of the solution's total curvatures) is imposed.
  3. [§3.2, proof of Theorem 3.9] The proof of the claim bounding T_i(τ s(t)+(1-τ) ŝ(t)) uses Lemma 3.13 again to assert |∂ ln T_{i,P}/∂s_j| ≤ C. This does follow from the sum estimate only because all cross-derivatives are negative by Lemma 3.4, but the manuscript does not spell out that step; more importantly, since Lemma 3.13 itself is not established, the entire weight estimate (3.7) and the subsequent comparison argument collapse. This reinforces the need to repair or replace Lemma 3.13 before the uniqueness theorem can be accepted.
minor comments (5)
  1. [§3.2, proof of Theorem 3.9] The references to "Theorem 3.11", "Theorem 3.12", and "Theorem 3.13" in the proof of Theorem 3.9 should be to Corollary 3.11, Lemma 3.12, and Lemma 3.13, respectively.
  2. [§3.1, Lemma 3.5] In the statement and proof of Lemma 3.5, the references to "Theorem 3.2 and Theorem 3.4" should be to Lemma 3.2 and Lemma 3.4, since those are the items containing the variational and monotonicity facts used there.
  3. [§3.2, Lemma 3.13 proof] In the proof of Lemma 3.13, the symbol T_i is used for the single-face quantity T_{i,P}; the subscript P should be kept throughout to avoid confusing the face contribution with the vertex total T_i defined in Definition 2.9.
  4. [§4, proofs of Theorems 1.4, 1.7, 1.9, 1.11] Several proofs refer to "Theorem 4.2" and "Theorem 4.3" when the cited statements are Lemma 4.2/Remark 4.2 and Lemma 4.3; this numbering should be corrected throughout Section 4.
  5. [§3.2, Lemma 3.13 case (1)] The inequality u^2/(1+u^2) ≤ C u arctan(1/u) for all u>0 is asserted without explicit constant; since the constant must be uniform across the face-boundedness hypotheses, the argument would benefit from a short derivation or an explicit choice of C.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the infinite-flow results are derived from independent finite-cell lemmas and a quoted maximum principle; no prediction reduces to its own input by construction.

full rationale

The paper's derivation chain is not circular. The flow (1.1) is defined from the difference T_i - T_hat_i, and the convergence theorems prove that the flow drives T_i toward T_hat_i through monotonicity and boundedness arguments; the conclusion is not assumed in the definition or in the side conditions. Theorem 1.4 and Theorem 1.7 use T_i(s0)-T_hat_i >= 0 or <= 0 only to obtain monotonicity, and the limits are then shown to satisfy the prescribed curvature equations. The finite building blocks (Lemma 2.3, Lemma 3.2, Lemma 3.4, Lemma 4.1, Lemma 4.3, Proposition 4.4) are quoted from [1] and [21]; although [21] is coauthored by the second author, those are separate finite-decomposition results whose assumptions do not include the present infinite well-posedness or convergence claims. The graph maximum principle used in the uniqueness proof is quoted from [13]; it is a general comparison principle, not a restatement of the target uniqueness theorem, and it does not itself assume the result being proved. There is no fitted parameter later renamed a prediction, no normalization that forces the conclusion, and no ansatz smuggled in through a citation: the generalized circle packing metric and total geodesic curvature formulas are taken from prior work as definitions, and the new mathematical content is the infinite flow argument built on them. The skeptical concern about Lemma 3.13 concerns a possible algebraic inconsistency with formula (2.1) that would affect the validity of the uniqueness proof; that is a correctness risk, not circularity, because the lemma is not assumed as an input but is asserted and proved inside the paper.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No data fitting parameters appear; the constants δ and ε in Theorem 1.11 are arbitrary positive inputs, not fitted. No new particles, forces, or geometric entities are postulated. The generalized circles (circles, horocycles, hypercycles) and the convention of treating a hypercycle axis as an abstract center are imported from [21]. The main axioms are borrowed theorems from [1], [13], and [21].

assumptions (6)
  • domain assumption For each polygon and positive curvatures a generalized circle packing exists and the dual curvature k_P is a C^1 function of the vertex curvatures.
    Quoted from [21, Lemma 2.3]; used to define generalized circle packing metrics and the total curvature functions T_i(s).
  • domain assumption The differential form ω = Σ T_i ds_i is closed on each polygon, giving symmetry ∂T_{i,P}/∂s_j = ∂T_{j,P}/∂s_i.
    Quoted from [1, Lemma 3.2]; used in maximum-principle and uniqueness computations.
  • domain assumption Derivative signs: ∂T_{i,P}/∂k_i > 0, ∂T_{i,P}/∂k_j < 0, and ∂(Σ_j T_{j,P})/∂s_i > 0.
    Quoted from [21, Lemma 3.4]; central to the maximum principle and to the signs of ω and h.
  • domain assumption Maximum principle II for infinite graphs with bounded weights (Lemma 3.10).
    Quoted from [13]; the engine of the uniqueness proof.
  • domain assumption Degenerate-circle limit laws: T_i → 0 as k_i → 0, and sums of T_i over degenerate vertices tend to |I|π or (n−2)π (Lemmas 4.1 and 4.3).
    Quoted from [21]; used to rule out s_i → −∞ in Theorem 1.4 and s_i → ∞ in Theorem 1.7.
  • standard math Arzelà-Ascoli and diagonal subsequence arguments justify passing from finite exhaustions to a global solution.
    Used in the proof of Theorem 3.1 to construct the infinite flow as a limit of flows on finite subcomplexes.

how reviews work

0 comments
Cite this review

Pith. "Pith review of A prescribed curvature flow on hyperbolic surfaces with infinite topological type." pith.science (2026). https://pith.science/paper/EP72I5QY

@misc{pith2026250520091,
  author       = {Pith},
  title        = {Pith review of: A prescribed curvature flow on hyperbolic surfaces with infinite topological type},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EP72I5QY}},
  note         = {Machine review of arXiv:2505.20091}
}
read the original abstract

In this paper, we investigate the prescribed total geodesic curvature problem for generalized circle packing metrics in hyperbolic background geometry on surfaces with infinite cellular decompositions. To address this problem, we introduce a prescribed curvature flow-a discrete analogue of the Ricci flow on noncompact surfaces-specifically adapted to the setting of infinite cellular decompositions. We establish the well-posedness of the flow and prove two convergence results under certain conditions. Our approach resolves the prescribed total geodesic curvature problem for a broad class of surfaces with infinite cellular decompositions, yielding, in certain cases, smooth hyperbolic surfaces of infinite topological type with geodesic boundaries or cusps. Moreover, the proposed flow provides a method for constructing hyperbolic metrics from appropriate initial data.

Figures

Figures reproduced from arXiv: 2505.20091 by the authors.

Figure 1
Figure 1. Circles, horocycles and hypercycles in the hyperbolic space [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. A generalized circle packing. We denote by ki the geodesic curvature of the circle Ci . We can classify the type of associated circle according to the value of geodesic curvature: (1) ki > 1, the associated circle is a hyperbolic circle. (2) ki = 1, the associated circle is a horocycle. (3) 0 < ki < 1, the associated circle is a hypercycle. The associated radius of Ci can be computed from its geodesic curvature as f… view at source ↗
Figure 3
Figure 3. A generalized pentagon of a generalized circle packing. [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: A generalized circle packing metric on a triangle that all circles are [PITH_FULL_IMAGE:figures/full_fig_p018_4.png]

Discussion (0). Sign in to comment.

Forward citations

Cited by 1 Pith paper

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

  1. Characterization of Infinite Ideal Polyhedra in Hyperbolic 3-Space via Combinatorial Ricci Flow

    math.GT 2025-06 reject novelty 6.0 of 10

    This paper proves new convergence results for infinite combinatorial Ricci flow on ideal circle patterns, but the claimed existence of infinite ideal hyperbolic polyhedra is not yet established.

Reference graph

Works this paper leans on

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

  1. [21]

    Hyperbolic circle packings and total geodesic curvatures on surfaces with boundary.Nonlinear Anal., 253:Paper No

    Guangming Hu, Yi Qi, Yu Sun, and Puchun Zhou. Hyperbolic circle packings and total geodesic curvatures on surfaces with boundary.Nonlinear Anal., 253:Paper No. 113735, 16, 2025

    work page 2025

  2. [13]

    Combinatorial ricci flows on infinite disk triangulations.arXiv preprint arXiv.2504.05817, 2025

    Huabin Ge, Bobo Hua, and Puchun Zhou. Combinatorial ricci flows on infinite disk triangulations.arXiv preprint arXiv.2504.05817, 2025

    arXiv 2025

  3. [1]

    Circle packings and hyperbolic surfaces of finite type

    Te Ba, Guangming Hu, and Yu Sun. Circle packings and total geodesic curva- tures in hyperbolic background geometry.arXiv preprint arXiv:2307.13572, 2023

    work page Pith review arXiv 2023

  4. [2]

    Bobenko and Boris A

    Alexander I. Bobenko and Boris A. Springborn. Variational principles for circle patterns and Koebe’s theorem.Trans. Amer. Math. Soc., 356(2):659– 689, 2004

    work page 2004

  5. [3]

    Manifolds with positive curvature operators are space forms.Ann

    Christoph B¨ ohm and Burkhard Wilking. Manifolds with positive curvature operators are space forms.Ann. of Math. (2), 167(3):1079–1097, 2008

    work page 2008

  6. [4]

    Kreispackungen und Triangulierungen.Enseign

    Walter Br¨ agger. Kreispackungen und Triangulierungen.Enseign. Math. (2), 38(3-4):201–217, 1992

    work page 1992

  7. [5]

    Manifolds with 1/4-pinched curvature are space forms.J

    Simon Brendle and Richard Schoen. Manifolds with 1/4-pinched curvature are space forms.J. Amer. Math. Soc., 22(1):287–307, 2009

    work page 2009

  8. [6]

    Strong uniqueness of the Ricci flow.J

    Bing-Long Chen. Strong uniqueness of the Ricci flow.J. Differential Geom., 82(2):363–382, 2009

    work page 2009

Show all 36 references
  1. [7]

    Uniqueness of the Ricci flow on complete noncompact manifolds.J

    Bing-Long Chen and Xi-Ping Zhu. Uniqueness of the Ricci flow on complete noncompact manifolds.J. Differential Geom., 74(1):119–154, 2006

    2006

  2. [8]

    The Ricci flow on the 2-sphere.J

    Bennett Chow. The Ricci flow on the 2-sphere.J. Differential Geom., 33(2):325–334, 1991

    1991

  3. [9]

    Combinatorial Ricci flows on surfaces.J

    Bennett Chow and Feng Luo. Combinatorial Ricci flows on surfaces.J. Differential Geom., 63(1):97–129, 2003

    2003

  4. [10]

    Colin de Verdi` ere

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

    1991

  5. [11]

    Combinatorial Ricci flows and the hyperbolization of a class of compact 3-manifolds.Geom

    Ke Feng, Huabin Ge, and Bobo Hua. Combinatorial Ricci flows and the hyperbolization of a class of compact 3-manifolds.Geom. Topol., 26(3):1349– 1384, 2022

    2022

  6. [12]

    Combinatorial methods and geometric equations.Ph.D

    Huabin Ge. Combinatorial methods and geometric equations.Ph.D. Thesis, Peking University, Beijing, 2012

    2012

  7. [14]

    Combinatorial Ricci flows for ideal circle patterns.Adv

    Huabin Ge, Bobo Hua, and Ze Zhou. Combinatorial Ricci flows for ideal circle patterns.Adv. Math., 383:Paper No. 107698, 26, 2021

    2021

  8. [15]

    Gregor Giesen and Peter M. Topping. Existence of Ricci flows of incomplete surfaces.Comm. Partial Differential Equations, 36(10):1860–1880, 2011

    2011

  9. [16]

    Hamilton

    Richard S. Hamilton. Three-manifolds with positive Ricci curvature.J. Dif- ferential Geometry, 17(2):255–306, 1982. 20

    1982

  10. [17]

    Hamilton

    Richard S. Hamilton. The Ricci flow on surfaces. InMathematics and general relativity (Santa Cruz, CA, 1986), volume 71 ofContemp. Math., pages 237–

    1986

  11. [18]

    Uniqueness and stability of singular Ricci flows in higher dimensions.Proc

    Robert Haslhofer. Uniqueness and stability of singular Ricci flows in higher dimensions.Proc. Amer. Math. Soc., 150(12):5433–5437, 2022

    2022

  12. [19]

    Rigidity of infinite disk patterns.Ann

    Zheng-Xu He. Rigidity of infinite disk patterns.Ann. of Math. (2), 149(1):1– 33, 1999

    1999

  13. [20]

    Zheng-Xu He and O. Schramm. Hyperbolic and parabolic packings.Discrete Comput. Geom., 14(2):123–149, 1995

    1995

  14. [22]

    Hirzel Stuttgart, 1936

    Paul Koebe.Kontaktprobleme der konformen Abbildung. Hirzel Stuttgart, 1936

    1936

  15. [23]

    On the uniqueness of Ricci flow.J

    Man-Chun Lee. On the uniqueness of Ricci flow.J. Geom. Anal., 29(4):3098– 3112, 2019

    2019

  16. [24]

    Characterizing the Delaunay decompositions of compact hyperbolic surfaces.Geom

    Gregory Leibon. Characterizing the Delaunay decompositions of compact hyperbolic surfaces.Geom. Topol., 6:361–391, 2002

    2002

  17. [25]

    Infinite combinatorial ricci flow in spherical background geometry.arxiv preprint arxiv.2505.05925, 2025

    Chang Li, Yangxiang Lu, and Hao Yu. Infinite combinatorial ricci flow in spherical background geometry.arxiv preprint arxiv.2505.05925, 2025

    2025 arXiv

  18. [26]

    Combinatorial Yamabe flow on surfaces.Commun

    Feng Luo. Combinatorial Yamabe flow on surfaces.Commun. Contemp. Math., 6(5):765–780, 2004

    2004

  19. [27]

    On circle patterns and spherical conical metrics.Proc

    Xin Nie. On circle patterns and spherical conical metrics.Proc. Amer. Math. Soc., 152(2):843–853, 2024

    2024

  20. [28]

    The entropy formula for the ricci flow and its geometric applications.arXiv preprint math/0211159, 2002

    Grisha Perelman. The entropy formula for the ricci flow and its geometric applications.arXiv preprint math/0211159, 2002

    2002 arXiv

  21. [29]

    Euclidean structures on simplicial surfaces and hyperbolic volume

    Igor Rivin. Euclidean structures on simplicial surfaces and hyperbolic volume. Ann. of Math. (2), 139(3):553–580, 1994

    1994

  22. [30]

    The convergence of circle packings to the Riemann mapping.J

    Burt Rodin and Dennis Sullivan. The convergence of circle packings to the Riemann mapping.J. Differential Geom., 26(2):349–360, 1987

    1987

  23. [31]

    Rigidity of infinite (circle) packings.J

    Oded Schramm. Rigidity of infinite (circle) packings.J. Amer. Math. Soc., 4(1):127–149, 1991

    1991

  24. [32]

    Deforming the metric on complete Riemannian manifolds

    Wan-Xiong Shi. Deforming the metric on complete Riemannian manifolds. J. Differential Geom., 30(1):223–301, 1989

    1989

  25. [33]

    W. P. Thurston. The geometry and topology of three-manifolds.Princeton lecture notes, 1976. 21

    1976

  26. [34]

    Peter M. Topping. Uniqueness of instantaneously complete Ricci flows.Geom. Topol., 19(3):1477–1492, 2015

    2015

  27. [35]

    Topping and Hao Yin

    Peter M. Topping and Hao Yin. Uniqueness of Ricci flows from non-atomic Radon measures on Riemann surfaces.Proc. Lond. Math. Soc. (3), 128(6):Pa- per No. e12600, 34, 2024. Xinrong Zhao, xrzhao24@m.fudan.edu.cn School of Mathematical Sciences, Fudan University, Shanghai, 200433...

    2024

  28. [262]

    Amer. Math. Soc., Providence, RI, 1988

    1988

Pith tools

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