Pith. sign in

REVIEW 3 major objections 5 minor 1 cited by

On the stability of Ricci flow on hyperbolic 3-manifolds of finite volume

T0 review · 3 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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

Pith's one-line read Cusped hyperbolic 3-manifolds are exponentially stable under normalized Ricci flow

desk verdict Cusp-shape deformations are neutral directions for the linearized flow, so the claimed spectral gap and exponential convergence to h0 cannot hold as stated. read the letter →

arxiv 2509.00188 v1 pith:AN5BFBBF submitted 2025-08-29 math.DG math.AP

Ruojing Jiang , Franco Vargas Pallete This is my paper · ORCID
classification math.DGmath.AP MSC 53E20
keywords Ricciflowhyperbolic3-manifoldscuspsstabilityinterpolationtheoryweightedHölderspacesDeTurckoperatorminimalsurfaceentropy
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 hyperbolic metric on a finite-volume hyperbolic 3-manifold, including its cusps, is dynamically stable under the normalized Ricci–DeTurck flow. Any smooth metric sufficiently close in the C0 norm flows for all time and converges exponentially fast to the hyperbolic metric in a weighted Hölder norm. This upgrades earlier qualitative convergence results, which only gave smooth convergence on compact sets, into a quantitative exponential statement. The rate is controlled by the spectral threshold −λ(2−λ) of the linearized DeTurck operator on the cusp. The proof rests on interpolation theory and weighted little Hölder spaces whose weights allow cuspidal growth while keeping the linearized operator surjective. If correct, the result supplies the exponential-convergence input needed for minimal-surface entropy comparison on finite-volume hyperbolic 3-manifolds.

What carries the argument

The central object is the family of weighted little Hölder spaces h^{k+ρ}_{λ,s}, defined with weight e^{−λr} in the cusps for λ∈(0,1) and (r+1)e^{−r} for λ=1. The paper uses K- and J-method interpolation theory, together with the Reiteration Theorem, to identify the intermediate spaces Xα=(X0,X1)α with h^{2α+ρ}_{λ,s}. The decisive computation is Lemma 6.2: averaging the linearized DeTurck equation over torus cross-sections of a cusp reduces it to the ODE system (6.4), whose characteristic roots select the L2-decaying modes and yield the spectral threshold ω0=−λ(2−λ). With ωI−Ah0 an isomorphism on the weighted spaces for Re ω>ω0, Angenent's maximal-regularity theorem makes Ah0 an admissible g

What would settle it

On the model cusp T²×[s,∞), compute the resolvent of ωI−Ah0 and check the threshold: solve the ODE system (6.4) with f=0 and look for nontrivial L2 solutions for ω>−λ(2−λ). If any exist, Lemma 6.2 and the rate in Theorem 1.1 are wrong; alternatively, a compactly supported f whose solution grows faster than the weight e^{−λr} would falsify the C0 estimate in Lemma 6.4.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.1: on any finite-volume hyperbolic 3-manifold (M,h0), for every λ∈(0,1] and every ω∈(0,λ(2−λ)), there exist ρ0,c>0 such that every smooth metric g with ∥g−h0∥C0<ρ0 evolves under the normalized Ricci–DeTurck flow (2.2) for all time, with ∥g(t)−h0∥X1 ≤ c(t−1)^{−(1−α)}e^{−ωt}∥g−h0∥C0 for t>1. The norm X1 is a weighted little Hölder space whose weight grows exponentially toward the cusps. The rate λ(2−λ) is exactly the gap coming from the linearized DeTurck operator on the cusp. This turns the earlier qualitative stability theorem, which gives convergence on compact subsets and preservation of the same asymptotic hyperbolic structure, into a quantitative exponentia

Load-bearing premise

The load-bearing premise is that the cusp-averaged linearized DeTurck equation reduces exactly to the ODE system (6.4) and that square-integrability kills the growing modes; if either fails, the claimed spectral threshold and exponential rate collapse.

Editorial extensions

If this is right

  • A C0-small perturbation of the hyperbolic metric never drifts away: the normalized Ricci–DeTurck flow exists globally and returns to h0 at an exponential rate controlled by the chosen weight.
  • The convergence holds in the full weighted Hölder norm X1, so geometric quantities built from the metric and its first two derivatives—such as curvature bounds—converge at the same exponential rate.
  • Because the rate can be taken arbitrarily close to λ(2−λ), the theorem provides a family of explicit decay rates rather than a single qualitative attractivity statement.
  • In the companion application [17], this exponential convergence is the input that yields E(h)≤E(h0) for weakly cusped metrics with scalar curvature ≥−6 on infinitesimally rigid finite-volume hyperbolic 3-manifolds, with equality only when h is isometric to h0.
  • The weighted-space setup is necessary: unweighted Hölder spaces fail because trivial Einstein variations in the cusps make the linearized operator non-surjective, while the opposite exponential weight would break the C1 regularity of the nonlinear operator.

Reading between the lines

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

  • A natural extension the paper does not pursue is whether the same weighted-space proof yields exponential stability for higher-dimensional finite-volume hyperbolic manifolds with rank-one cusps; the ODE system would change, but the interpolation framework should carry over.
  • The explicit rate λ(2−λ) suggests a practical trade-off: choosing larger λ gives faster decay but imposes a heavier weight, so an optimal choice would depend on the perturbation class being measured.
  • Because the theorem controls weighted Hölder norms all the way to the cusp, it should be usable as a quantitative convergence certificate for numerical Ricci-flow computations on noncompact manifolds.
  • A sharper version might allow initial perturbations that grow slowly toward the cusp under a weighted C0 norm; Remark 5.4 hints that the only excluded directions are trivial Einstein variations, which are precisely what the weight enforces.
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 proves a quantitative stability theorem for the normalized Ricci-DeTurck flow near the hyperbolic metric on a finite-volume hyperbolic 3-manifold. It introduces exponentially weighted little Hölder spaces h^{k+α}_{λ,s} and shows (Theorem 1.1) that any smooth metric sufficiently C^0-close to h0 produces a global flow converging to h0 in X1 = h^{2+ρ}_{λ,s} with exponential rate e^{-ωt} for every ω < λ(2−λ). The proof follows the Da Prato–Grisvard/Angenent maximal-regularity framework: the linearized DeTurck operator is analyzed on weighted spaces, a cusp-averaged ODE system is used to establish the resolvent threshold ω0 = -λ(2−λ), Schauder and L^2 estimates give injectivity, Lax–Milgram gives surjectivity, and a Duhamel formula with interpolation estimates yields the exponential decay. The paper also advertises an application to minimal surface entropy in a companion article [17].

Significance. If correct, this is a real advance: it upgrades Bamler's qualitative C^0/C^k stability for cusped hyperbolic 3-manifolds to quantitative exponential convergence, with a decay rate derived from the spectral analysis of the linearized DeTurck operator rather than from fitted parameters. The weighted-space setup is well motivated by the presence of trivial Einstein variations in the cusps, and the theorem is stated uniformly for all admissible weights λ∈(0,1], which strengthens the claim. The proof strategy is natural and the paper is clearly organized. The principal caveat is that several load-bearing computations are imported from an external preprint [12] and the mode-selection step in the cusp ODE analysis is not fully self-contained; independent verifiability of these steps is essential because the claimed spectral threshold drives the final exponential rate.

major comments (3)
  1. [Section 6.2, Eq. (6.4)] The central spectral computation is not self-contained. The reduction of (ωI − A_{h0})l = f to the ODE system (6.4) is stated as 'calculated in (9.14) of [12]', and the threshold ω0 = -λ(2−λ) in Lemma 6.2, the resolvent estimates in Proposition 6.1, and every admissible rate in Theorem 1.1 depend on this system. Please provide a complete derivation, including the cusp coordinate/frame convention, the precise definition of the averaged components l̂_ij, and the origin of the constants 1, 3, 4 and of the trace-coupling term 2δ_ij(tr l̂ − l̂_33) in (6.4). Without this, the main theorem cannot be independently checked by the reader.
  2. [Section 6.2.2, Lemma 6.2 (mode selection)] The square-integrability step after (6.7) is too compressed. The assertion that e^{-r}(e^{2r}l̂_12), e^{-r}(e^{r}l̂_i3), e^{-r}l̂_33, e^{-r}tr(l̂) ∈ L^2, together with 'any root with real part ≥ 1 is not square integrable', must be matched exactly with the tensor norm and volume element used in the weighted Hölder spaces. The admissible exponents 1−√(1+ω) can be positive when Re ω is near ω0, so the conclusion is not a purely formal square-integrability statement in coordinate components. The paper should prove explicitly that every solution l∈E1 (not only l∈H^1) satisfies the mode-selection criterion, and should verify that the particular integrals in (6.7) remain O(∥f∥_{E0} w_λ^{-1}) uniformly as Re ω ↓ ω0, with no resonance loss. This step is load-bearing for the resolvent threshold.
  3. [Section 6.2.4, Proposition 6.6] The uniform resolvent bound is necessary to conclude that A_{h0} generates a strongly continuous analytic semigroup on (E0,E1), but the proof by contradiction is incomplete. In Case 1, from ∥L_n∥_{E0}→0 and ∥L_n∥_{E1}≤1 it does not follow directly that L_n→0 locally in E1; one needs a local Schauder estimate on the shifted cusp domains. In Case 2, the statement that 'all dependencies on ω used in the proofs of Lemma 6.2, Lemma 6.4 and Corollary 6.5 can be uniformly controlled' is asserted without details. Please provide a rigorous compactness/Schauder argument or an alternative sectoriality proof. This uniformity is essential for condition (C3).
minor comments (5)
  1. [Theorem 1.1] In the statement, the smallness condition is written as ∥h−h0∥_{C^0(M)}, but the metric is called g; it should read ∥g−h0∥_{C^0(M)}. Also 'There exist ρ0, c >0,' should be 'there exist'.
  2. [Section 7, displayed linear system] The linear system is written as ∂_t H = A_{h0}H + (A(h(t))−A_{h0})H, but the Duhamel formula immediately below uses h(s) in the integrand. Since h(t) is then identified with H(t), the intended inhomogeneous linear equation is ∂_t H = A_{h0}H + (A(h(t))−A_{h0})h(t), and the variation-of-constants formula should consistently use h(s). Please correct this internal inconsistency.
  3. [End of Section 7] The final estimate writes ∥l(0)∥_{C^2(M)} on the right-hand side, while Theorem 1.1 and the preceding norms use the Xα norm and then the C^0 norm of g−h0 via Theorem 2.1. Please make the norm chain explicit: Xα ≤ C∥g(1)−h0∥_{C^3} ≤ C∥g(0)−h0∥_{C^0}.
  4. [Section 6.2.2, after (6.15)] The phrase 'where ρ is a universal constant' appears after (6.15), but ρ is not defined in that estimate; the displayed inequality contains no ρ. Either remove the phrase or define the intended radius.
  5. [Throughout] Several small typos occur, e.g. 'Cauchy-Schwartz' should be 'Cauchy-Schwarz'. More importantly, the paper never explicitly writes the cusp metric model (e.g. dr^2 + e^{-2r}g_T or the opposite convention); adding this model at the start of Section 5 would clarify the weight, the volume element, and the ODE system in Lemma 6.2.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the exponential rate is derived from spectral analysis of the linearized DeTurck operator; the only self-citation, [17], is not load-bearing.

full rationale

The paper's central claim does not reduce to its inputs by construction. The exponential decay rate in Theorem 1.1 is not a fitted parameter: the theorem is proved for every λ∈(0,1], and for every ω<λ(2−λ), with the spectral threshold ω0=−λ(2−λ) computed in Lemma 6.2 from the characteristic roots of the cusp-averaged ODE system. The weighted little Hölder spaces are defined before the theorem, and the theorem holds uniformly across the family of norms, so the rate is not an artifact of tuning a weight to force the conclusion. The only self-citation is [17], a companion paper in preparation, whose Theorems 1.2–1.3 are explicitly deferred and not used in the proof of Theorem 1.1. The spectral reduction in Lemma 6.2 imports the ODE system (6.4) from Hamenstädt–Jäckel [12, (9.14)], which is an external prior work, not a self-citation; any concern about the correctness of that reduction is a verification risk, not circularity. Likewise, Bamler's stability theorem [5] supplies long-time existence and C^k closeness as an independent external input. No equation in the paper is shown to be equivalent by construction to the claimed conclusion, and no fitted parameter is renamed as a prediction. Accordingly, no significant circularity is present.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new physical entities. The weighted Hölder spaces h^{k+ρ}_{λ,s} are definitions (function-space constructions), not postulates with independent falsifiable handles; they are captured under free parameters and axioms.

free parameters (4)
  • weight exponent λ = any value in (0,1]
    Defines the weighted Hölder norm w_λ=e^{-λr}; the spectral threshold ω0=-λ(2-λ) depends on λ, and the theorem quantifies convergence for any λ. It is a norm choice, not fitted to data.
  • cusp cutoff s = any s>0
    Defines the thick-thin decomposition M(s); constants in the proof may depend on s, but the conclusion holds for the chosen s.
  • Hölder exponents ρ, σ = 0<σ<ρ<1
    Parameters of the spaces X0=h^{0+ρ}, X1=h^{2+ρ}, E0=h^{0+σ}, E1=h^{2+σ}; the interpolation index θ=(ρ−σ)/2 in condition (C4) uses them.
  • interpolation index α = α∈(0,1), α≠(1−ρ)/2, 1−ρ/2
    The theorem's decay has the factor t^{-(1−α)}; the interpolation identity Xα=(X0,X1)_α=h^{2α+ρ} requires 2α+ρ∉N.
assumptions (6)
  • standard math Real interpolation theory: K-method, J-method, Reiteration Theorem (Theorem 3.4)
    Used throughout to identify Xα and for Corollary 5.3; cited to Lunardi [26] and Triebel [36].
  • standard math Simonett's stability theorem (Theorem 4.1) and Angenent's maximal regularity theorem (Theorem 4.2)
    The abstract framework for quasilinear parabolic equations and linear maximal regularity; quoted from [33] and [3].
  • domain assumption Bamler's stability theorem: long-time existence and C^k bounds for small C^0 perturbations of cusped hyperbolic metrics (Theorem 2.1)
    Provides the a priori C^k control of the nonlinear flow used to bound the nonlinearity in Section 7; taken from [5].
  • domain assumption Hamenstädt-Jäckel estimates: cusp ODE system (9.14), Lemma 9.21, Proposition 3.1 Poincaré inequality, Corollary 7.7, and DGN argument
    The core spectral and elliptic estimates in Section 6 are adapted from [12]; the paper relies on these without reproducing them.
  • domain assumption Finite-volume hyperbolic 3-manifold structure: torus cusps, thick-thin decomposition, universal cover H^3
    The geometry of M underpins the weight r(x), the ODE model and the lifting arguments throughout Sections 5-6.
  • standard math Elliptic regularity: Schauder estimates, Lax-Milgram, Weyl lemma, De Giorgi-Nash-Moser
    Used in Sections 6.2.1 and 6.2.3 to obtain coercivity, regularity and pointwise bounds.

how reviews work

0 comments
Cite this review

Pith. "Pith review of On the stability of Ricci flow on hyperbolic 3-manifolds of finite volume." pith.science (2026). https://pith.science/paper/AN5BFBBF

@misc{pith2026250900188,
  author       = {Pith},
  title        = {Pith review of: On the stability of Ricci flow on hyperbolic 3-manifolds of finite volume},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AN5BFBBF}},
  note         = {Machine review of arXiv:2509.00188}
}
abstract

On a hyperbolic 3-manifold of finite volume, we prove that if the initial metric is sufficiently close to the hyperbolic metric $h_0$, then the normalized Ricci-DeTurck flow exists for all time and converges exponentially fast to $h_0$ in a weighted H\"older norm. A key ingredient of our approach is the application of interpolation theory. Furthermore, this result is a valuable tool for investigating minimal surface entropy, which quantifies the growth rate of the number of closed minimal surfaces in terms of genus. We explore this in [17].

Discussion (0). Continue with ORCID 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. Volume comparison on finite-volume hyperbolic 3-manifolds

    math.DG 2025-09 conditional novelty 7.0 of 10

    The hyperbolic metric minimizes volume among all metrics with scalar curvature at least -6 on finite-volume hyperbolic 3-manifolds, with rigidity when the metric is C^2-close or asymptotically cusped.

Reference graph

Works this paper leans on

38 extracted references · 1 canonical work pages · cited by 1 Pith paper

  1. [12]

    Hamenstädt and F

    U. Hamenstädt and F. Jäckel. Stability of Einstein metrics and effective hyperbolization in large Hempel distance.arXiv:2206.10438, 2022

    arXiv 2022

  2. [17]

    Jiang and F

    R. Jiang and F. Vargas Pallete. Minimal surface entropy and applications of Ricci flow on finite-volume hyperbolic 3-manifolds, 2025. Manuscript in preparation

    2025

  3. [1]

    H. Amann. Linear and Quasilinear Parabolic Problems. Volume I: Abstract Linear Theory. Monographs in Mathematics. Birkhäuser Basel, 1995

    1995

  4. [2]

    Anderson

    M.T. Anderson. Dehn filling and Einstein metrics in higher dimensions.J. Differential Geom., 73(2):219–261, 2006

    2006

  5. [3]

    Angenent

    S.B. Angenent. Nonlinear analytic semiflows.Proc. Roy. Soc. Edinburgh, 115(1-2):91–107, 1990

    1990

  6. [4]

    R.H. Bamler. Construction of Einstein metrics by generalized Dehn filling.J. Eur. Math. Soc., 14(3):887–909, 2012

    2012

  7. [5]

    R.H. Bamler. Stability of hyperbolic manifolds with cusps under Ricci flow. Adv. Math., 263:412–467, 2014

    2014

  8. [6]

    Bessières, G

    L. Bessières, G. Besson, and S. Maillot. Long time behaviour of Ricci flow on open 3-manifolds. Comment. Math., 90(2):377–405, 2015

    2015

Show all 38 references
  1. [7]

    Calegari, F

    D. Calegari, F. Marques, and A. Neves. Counting minimal surfaces in negatively curved 3- manifolds. Duke Math. J., 2022

    2022

  2. [8]

    Gilbarg and N.S

    D. Gilbarg and N.S. Trudinger.Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer Berlin, Heidelberg, 2001

    2001

  3. [9]

    M. Gromov. Metric Structures for Riemannian and Non-Riemannian Spaces . Modern Birkhäuser Classics. Birkhäuser Boston, reprint of the 1999 english edition edition, 2007. Trans- lated from the French by Sean Michael Bates, with appendices by M. Katz, P. Pansu, and S. Semmes

    1999

  4. [10]

    Guenther, I

    C. Guenther, I. Isenberg, and D. Knopf. Stability of the Ricci flow at Ricci-flat metrics.Comm. Anal. Geom., 10(4):741–777, 2002

    2002

  5. [11]

    Hamenstädt

    U. Hamenstädt. Entropy-rigidity of locally symmetric spaces of negative curvature.Ann. of Math., 131(1):35–51, 1990

    1990

  6. [13]

    R. S. Hamilton. Three-manifolds with positive Ricci curvature.J. Differential Geom., 17:255– 306, 1982

    1982

  7. [14]

    Hebey.Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities

    E. Hebey.Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities. Courant lecture notes in mathematics. Courant Institute of Mathematical Sciences, 1999

    1999

  8. [15]

    B. Hu, L. Ji, and Y. Shi. Stability of conformally compact Einstein manifolds.J. Funct. Anal., 278(12):108455, 2020

    2020

  9. [16]

    L. Ji, R. Mazzeo, and N. Šešum. Ricci flow on surfaces with cusps.Math. Ann., 345(4):819–834, 2009

    2009

  10. [18]

    Kahn and V

    J. Kahn and V. Markovic. Counting essential surfaces in a closed hyperbolic 3-manifold.Geom. Topol., 16:601–624, 2012

    2012

  11. [19]

    Kahn and V

    J. Kahn and V. Markovic. Immersing almost geodesic surfaces in a closed hyperbolic three manifold. Ann. of Math., 175(3):1127–1190, 2012. STABILITY OF RICCI FLOW ON HYPERBOLIC 3-MANIFOLDS OF FINITE VOLUME 35

    2012

  12. [20]

    D. Knopf. Convergence and stability of locallyRN-invariant solutions of Ricci flow.J. Geom. Anal., 19:817–846, 2009

    2009

  13. [21]

    Knopf and A

    D. Knopf and A. Young. Asymptotic stability of the cross curvature flow at a hyperbolic metric. Proc. Amer. Math. Soc., 137(2):699–709, 2009

    2009

  14. [22]

    Li and H

    H. Li and H. Yin. On the stability of the hyperbolic space form under the normalized Ricci flow. Int. Math. Res. Not., 2010(15):2903–2924, 2010

    2010

  15. [23]

    Lions and J

    J.-L. Lions and J. Peetre. Sur une classe d’espaces d’interpolation.Publications Mathématiques de l’IHÉS, 19:5–68, 1964

    1964

  16. [24]

    Lowe and A

    B. Lowe and A. Neves. Minimal surface entropy and average area ratio.J. Differential Geom., 130(2):443–475, 2025

    2025

  17. [25]

    A. Lunardi. Analytic Semigroups and Optimal Regularity in Parabolic Problems. Modern Birkhäuser Classics. Springer Basel, 1995

    1995

  18. [26]

    Lunardi.Interpolation Theory

    A. Lunardi.Interpolation Theory. Publications of the Scuola Normale Superiore. Edizioni della Normale Pisa, 2018

    2018

  19. [27]

    Perelman

    G. Perelman. The entropy formula for the Ricci flow and its geometric applications, 2002

    2002

  20. [28]

    G.Perelman.FiniteextinctiontimeforthesolutionstotheRicciflowoncertainthree-manifolds, 2003

    2003

  21. [29]

    Perelman

    G. Perelman. Ricci flow with surgery on three-manifolds, 2003

    2003

  22. [30]

    Annali della Scuola Normale Superiore di Pisa, 2:311–393, 1975

    G.DaPratoandP.Grisvard.Equationsd’évolutionabstraitesnonlinéairesdetypeparabolique. Annali della Scuola Normale Superiore di Pisa, 2:311–393, 1975

    1975

  23. [31]

    O. C. Schnürer, F. Schulze, and M. Simon. Stability of Euclidean space under Ricci flow.Comm. Anal. Geom., 16(1):127–158, 2008

    2008

  24. [32]

    N. Šešum. Linear and dynamical stability of Ricci-flat metrics.Duke Math. J., 133(1):1–26, 2006

    2006

  25. [33]

    Simonett

    G. Simonett. Center manifolds for quasilinear reaction-diffusion systems.Differential and Inte- gral Equations, 8(4):753–796, 1995

    1995

  26. [34]

    G. Tian. A pinching theorem on manifolds with negative curvature.Proceedings of International Conference on Algebraic and Analytic Geometry, 1990

    1990

  27. [35]

    Topping.Lectures on the Ricci Flow

    P. Topping.Lectures on the Ricci Flow. London Mathematical Society Lecture Note Series. Cambridge University Press, 2006

    2006

  28. [36]

    Triebel.Interpolation Theory, Function Spaces, Differential Operators

    H. Triebel.Interpolation Theory, Function Spaces, Differential Operators. 2., rev. and enl. ed. Johann Ambrosius Barth Verlag, Heidelberg, 1995

    1995

  29. [37]

    H. Wu. Stability of complex hyperbolic space under curvature-normalized Ricci flow.Geome- triae Dedicata, 164:231–258, 2013

    2013

  30. [38]

    R. Ye. Ricci flow, Einstein metrics and space forms.Trans. Amer. Math. Soc., 338(2):871–896, 1993. Massachusetts Institute of Technology, Department of Mathematics, Cambridge, MA 02139 Email address: ruojingj@mit.edu School of Mathematical and Statistical Sciences, Arizona Sta...

    1993

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.