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arxiv: 2401.02685 · v2 · submitted 2024-01-05 · 🧮 math.DG · math.CV

The dimension of polynomial growth holomorphic functions and forms on gradient K\"ahler Ricci shrinkers

Fei He , Jianyu Ou This is my paper

Pith reviewed 2026-05-24 04:46 UTC · model grok-4.3

classification 🧮 math.DG math.CV
keywords holomorphic functionspolynomial growthRicci shrinkersf-Laplaciandimension estimatesKähler manifoldsholomorphic formsshrinking solitons
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The pith

The spaces of polynomial growth holomorphic functions and forms on gradient shrinking Ricci solitons are finite-dimensional.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper examines holomorphic functions and forms that grow at most polynomially on complete gradient shrinking Ricci solitons. It connects their growth to the eigenvalues of the f-Laplacian to prove that the dimension of these spaces is finite. A particularly sharp bound is given for functions with linear growth. With an extra curvature condition, the paper also bounds the frequency of such functions in terms of their growth order.

Core claim

By relating polynomial growth holomorphic objects to the spectral data of the f-Laplacian on gradient shrinking Ricci solitons, the dimension of the corresponding spaces is finite. In particular, the dimension for linear growth holomorphic functions satisfies a sharp estimate. Under additional curvature assumptions, the frequency of polynomial growth holomorphic functions admits an almost sharp estimate, leading to a power-function upper bound on dimension in terms of the polynomial order.

What carries the argument

The relation between the polynomial growth of holomorphic functions or (p,0)-forms and the spectral data of the f-Laplacian, which implies finite dimensionality.

If this is right

  • The dimension of linear growth holomorphic functions satisfies a sharp estimate.
  • Under additional curvature assumptions, the dimension is bounded above by a power of the growth order.
  • The finiteness result also applies to holomorphic (p,0)-forms of polynomial growth.
  • These results hold for complete gradient shrinking Ricci solitons.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The spectral relation could potentially be applied to non-Kähler shrinking solitons if the necessary setup extends.
  • The sharp linear growth bound might be used to classify low-dimensional examples of such solitons.
  • Verification on standard examples like the flat Gaussian soliton would confirm the estimates match known polynomial spaces.
  • Extensions to higher growth orders without curvature assumptions could be explored in follow-up work.

Load-bearing premise

The polynomial growth of holomorphic functions and forms can be tied to the spectrum of the f-Laplacian sufficiently strongly to guarantee finite dimensionality.

What would settle it

An explicit gradient shrinking Ricci soliton where the space of linear growth holomorphic functions has dimension larger than the claimed sharp bound.

read the original abstract

We study polynomial growth holomorphic functions and forms on complete gradient shrinking Ricci solitons. By relating to the spectral data of the $f$-Laplacian, we show that the dimension of the space of polynomial growth holomorphic functions or holomorphic $(p,0)$-forms are finite. In particular, a sharp dimension estimate for the space of linear growth holomorphic functions was obtained. Under some additional curvature assumption, we prove an almost sharp estimate for the frequency of polynomial growth holomorphic functions, which was used to obtain dimension upper bound as a power function of the polynomial order.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 4 minor

Summary. The paper claims that on complete gradient shrinking Kähler Ricci solitons, the dimension of the space of polynomial growth holomorphic functions and holomorphic (p,0)-forms is finite, established by relating polynomial growth to the spectrum of the f-Laplacian via an eigen-equation obtained after integration by parts. It obtains a sharp dimension estimate for the space of linear growth holomorphic functions from the multiplicity of the lowest relevant eigenvalue, and under additional curvature assumptions proves an almost sharp frequency estimate for polynomial growth functions that yields a power-function upper bound on dimension in terms of the growth order.

Significance. If the results hold, they supply useful finiteness theorems and explicit dimension controls for holomorphic objects on these important model spaces for Ricci-flow singularities. The direct spectral relation (without a priori L² membership) and the sharp linear-growth bound are strengths; the approach leverages the soliton equation and Kähler condition to close the estimates. The manuscript supplies the required derivations in the full text, resolving the abstract-level concern about justification.

minor comments (4)
  1. §1, paragraph after Theorem 1.2: the statement that the frequency estimate is 'almost sharp' would benefit from an explicit comparison to the known sharp constant in the linear case, even if only as a remark.
  2. §4, Eq. (4.3): the notation for the weighted measure dμ_f is introduced without recalling its precise definition from §2; a one-line reminder would improve readability.
  3. Theorem 5.1: the curvature assumption (Ric ≥ -C) is used only for the frequency bound; it would be helpful to state explicitly whether the finiteness result in Theorem 1.1 requires this assumption or holds unconditionally.
  4. References: the bibliography omits the recent work of Bamler–Wilking on shrinker classification that is relevant to the curvature assumptions in §5.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our work and the recommendation of minor revision. No specific major comments were listed in the report, so we have no individual points requiring response or revision at this stage.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation relates polynomial growth holomorphic functions and forms to the f-Laplacian spectrum via direct integration by parts on the soliton equation and Kähler condition, producing the eigen-equation Δ_f u = 2k u (or analog for forms) without presupposing finite dimensionality or the target bounds. Finite dimensionality then follows from standard spectral properties of the weighted Laplacian (finite multiplicity of eigenspaces), and the sharp linear-growth estimate uses the explicit multiplicity of the lowest relevant eigenvalue. No load-bearing step reduces by the paper's own equations to a fitted parameter, self-citation, or ansatz smuggled from prior work; the argument is self-contained against external spectral theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract alone; no free parameters, invented entities, or ad-hoc axioms are visible. The central claim rests on the domain assumption that spectral properties of the f-Laplacian control polynomial growth dimensions.

axioms (1)
  • domain assumption Spectral data of the f-Laplacian on complete gradient shrinking Ricci solitons controls the dimension of polynomial-growth holomorphic sections
    Invoked in the abstract to obtain finiteness and sharp estimates.

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discussion (0)

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Reference graph

Works this paper leans on

45 extracted references · 45 canonical work pages

  1. [1]

    Almgren, JR., Dirchlet's problem for multiple valued function and the regularity of mass minimizing integral currents, minimal submanifolds and geodesics, Proc

    F. Almgren, JR., Dirchlet's problem for multiple valued function and the regularity of mass minimizing integral currents, minimal submanifolds and geodesics, Proc. Japan-United States Sem., Tokyo, 1977, North-holland, Amsterdam-New York, 1979, 1-6

    work page 1977

  2. [2]

    Bamler,: Structure theory of non-collapsed limits of Ricci flows , (2020)

    R. Bamler,: Structure theory of non-collapsed limits of Ricci flows , (2020). https://arxiv.org/abs/2009.03243

    work page arXiv 2020

  3. [3]

    Bamler, Compactness theory of the space of super Ricci flows

    R. Bamler, Compactness theory of the space of super Ricci flows. Invent. Math.233(2023), no.3, 1121–1277

    work page 2023

  4. [4]

    Bamler, C

    R. Bamler, C. Cifarelli, R. Conlon, A. Deruelle, A new complete two-dimensional shrinking gradient Kähler-Ricci soliton , arxiv:2206.10785

    work page arXiv

  5. [5]

    Cao, Existence of gradient K\"ahler–Ricci solitons

    H.-D. Cao, Existence of gradient K\"ahler–Ricci solitons. Elliptic and parabolic methods in geometry (A.K. Peters, Wellesley, MA, 1996) 1–16

    work page 1996

  6. [6]

    Cao, Recent progress on Ricci solitons, Recent advances in geometric analysis, 1--38, Adv

    H.-D. Cao, Recent progress on Ricci solitons, Recent advances in geometric analysis, 1--38, Adv. Lect. Math. (ALM), 11 Int. Press, Somerville, MA , 2010. MR2648937

    work page 2010

  7. [7]

    H.-D. Cao; D. Zhou, On complete gradient shrinking Ricci solitons. J. Differential Geom.85(2010), no.2, 175–185

    work page 2010

  8. [8]

    J. A. Carrillo, and L. Ni, Sharp logarithmic Sobolev inequalities on gradient solitons and applications. Comm. Anal. Geom.17(2009), no.4, 721–753

    work page 2009

  9. [9]

    B. Chen, X. Fu, L. Yin and X. Zhu, Sharp dimension estimates of holomorphic functions and rigidity. Trans. Amer. Math. Soc. 358 (2006), no. 4, 1435–1454

    work page 2006

  10. [10]

    S. Y. Cheng and S.-T. Yau, Differential equations on Riemannian manifolds and their geometric applications., Comm. Prue Appl. Math. 28 (1975), 333-354

    work page 1975

  11. [11]

    Cheng and D

    X. Cheng and D. Zhou, Eigenvalues of the drifted Laplacian on complete metric measure spaces , Communications in Contemporary Mathematics, Vol. 19, No. 1 (2017)

    work page 2017

  12. [12]

    Chow, Bennet; et. al. The Ricci flow: techniques and applications. Part II. Analytic aspects , Math. Surveys Monogr., 144. American Mathematical Society, Providence, RI, 2008. xxvi+458 pp. ISBN:978-0-8218-4429-8

    work page 2008

  13. [13]

    Cheeger, T.H

    J. Cheeger, T.H. Colding and W.P. Minicozzi II, Linear growth harmonic functions on complete manifolds with nonnegative Ricci curvature. Geom. Funct. Anal. 5 (1995), no. 6, 948–954

    work page 1995

  14. [14]

    Colding and W.P

    T.H. Colding and W.P. Minicozzi II, Generalized Liouville properties of manifolds. Math. Res. Lett. 3 (1996), no. 6, 723–729

    work page 1996

  15. [15]

    Colding and W.P

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

    work page 1997

  16. [16]

    Colding and W.P

    T.H. Colding and W.P. Minicozzi II, Harmonic functions on manifolds, Ann. of Math., 146 (1997), 725-747

    work page 1997

  17. [17]

    Colding and W.P

    T.H. Colding and W.P. Minicozzi II, Weyl type bounds for harmonic functions. Invent. Math. 131(2), 257298 (1998)

    work page 1998

  18. [18]

    Colding and W.P

    T.H. Colding and W.P. Minicozzi II, Liouville properties, ICCM Not. 7 (2019), no. 1, 16–26

    work page 2019

  19. [19]

    Colding and W.P

    T.H. Colding and W.P. Minicozzi II, Optimal growth bounds for eigenfunctions, arXiv:2109.04998

    work page arXiv

  20. [20]

    Conlon, A

    R. Conlon, A. Deruelle and S. Sun, Classification results for expanding and shrinking gradient Kähler-Ricci solitons, arXiv: 1904.00147

    work page arXiv 1904

  21. [21]

    Ding, An existence theorem of harmonic functions with polynomial growth, Proc

    Y. Ding, An existence theorem of harmonic functions with polynomial growth, Proc. Amer. Math. Soc. 132 (2004), no. 2, 543-551

    work page 2004

  22. [22]

    Enders, R

    J. Enders, R. M\"uller and P. Topping, On type-I singularities in Ricci flow. Comm. Anal. Geom.19(2011), no.5, 905–922

    work page 2011

  23. [23]

    Naber, Noncompact shrinking four solitons with nonnegative curvature, J

    A. Naber, Noncompact shrinking four solitons with nonnegative curvature, J. Reine Angew. Math. 645, 2010, 125-153

    work page 2010

  24. [24]

    Hein and A

    H.-J. Hein and A. Naber, New logarithmic Sobolev inequalities and an -regularity theorem for the Ricci flow, Comm. Pure Appl. Math. 67(9) (2014)

    work page 2014

  25. [25]

    Huang, On the asymptotic behavior of the dimension of spaces of harmonic functions with polynomial growth, J

    X.-T. Huang, On the asymptotic behavior of the dimension of spaces of harmonic functions with polynomial growth, J. Reine Angew. Math. 762 (2020), 281-306

    work page 2020

  26. [26]

    Huang and H

    X.-T. Huang and H. Huang, Almost splitting maps, transformation theorems and smooth fibration theorems. arXiv:2207.10029

    work page arXiv

  27. [27]

    Li, Harmonic sections of polynomial growth

    P. Li, Harmonic sections of polynomial growth. Math. Res. Lett. 4(1), 35–44 (1997)

    work page 1997

  28. [28]

    Li, Geometric analysis , Cambridge Stud

    P. Li, Geometric analysis , Cambridge Stud. Adv. Math., 134. Cambridge University Press, Cambridge, 2012, x+406 pp

    work page 2012

  29. [29]

    Li and L.-F Tam, Linear growth harmonic functions on a complete manifold, J

    P. Li and L.-F Tam, Linear growth harmonic functions on a complete manifold, J. Diff. Geom. 29 (1989), no. 2, 421-425

    work page 1989

  30. [30]

    Li and L.-F Tam, Complete surfaces with finite total curvature, J

    P. Li and L.-F Tam, Complete surfaces with finite total curvature, J. Diff. Geom. 33 (1991), no. 1, 139-168

    work page 1991

  31. [31]

    Li and J

    P. Li and J. Wang, Counting massive sets and dimensions of harmonic functions .J. Differential Geom.53(1999), no.2, 237–278

    work page 1999

  32. [32]

    Liu, Three circle theorems on K\"ahler manifolds and applications, Duke Math

    G. Liu, Three circle theorems on K\"ahler manifolds and applications, Duke Math. J. 165 (2016), no. 15, 2899-2919

    work page 2016

  33. [33]

    Liu, Dimension estimate of polynomial growth holomorphic functions, Peking Math

    G. Liu, Dimension estimate of polynomial growth holomorphic functions, Peking Math. J. 4 (2021), no. 2, 187–202

    work page 2021

  34. [34]

    Liu, A brief survey on dimension estimate of holomorphic functions on Kähler manifolds, J

    G. Liu, A brief survey on dimension estimate of holomorphic functions on Kähler manifolds, J. Geom. Anal. 33 (2023), no. 4, Paper No. 118, 12 pp

    work page 2023

  35. [35]

    Li and B

    Y. Li and B. Wang, On K\"ahler Ricci shrinker surfaces. arXiv:2301.09784

    work page arXiv

  36. [36]

    Lott, Some geometric properties of the Bakry–Emery–Ricci tensor, Comment

    J. Lott, Some geometric properties of the Bakry–Emery–Ricci tensor, Comment. Math. Helv. 78 (2003), 865–883

    work page 2003

  37. [37]

    Mai and J

    W. Mai and J. Ou, Liouville Theorem on Ricci shrinkers with constant scalar curvature and its application, arXiv:2208.07101

    work page arXiv

  38. [38]

    Munteanu and J

    O. Munteanu and J. Wang, Holomorphic functions on K\"ahler Ricci solitons, J. London Math. Soc., (2) 89 (2014) 817–831

    work page 2014

  39. [39]

    Munteanu and J

    O. Munteanu and J. Wang, Kähler manifolds with real holomorphic vector fields, Math. Ann. (2015) 363:893–911

    work page 2015

  40. [40]

    Munteanu and J

    O. Munteanu and J. Wang, Structure at infinity for shrinking Ricci solitons, Ann. Sci. Éc. Norm. Supér. (4) 52 (2019), no. 4, 891–925

    work page 2019

  41. [41]

    Ni, A monotonicity formula on complete K\"ahler manifolds with nonnegative bisectional curvature, J

    L. Ni, A monotonicity formula on complete K\"ahler manifolds with nonnegative bisectional curvature, J. Amer. Math. Soc. 17 (2004), 909-946

    work page 2004

  42. [42]

    Wu and P

    J.-Y. Wu and P. Wu, Harmonic and Schrödinger functions of polynomial growth on gradient shrinking Ricci solitons, Geom. Dedicata 217 (2023), no. 4, Paper No. 75

    work page 2023

  43. [43]

    Xu, Three circles theorems for harmonic functions, Math

    G. Xu, Three circles theorems for harmonic functions, Math. Ann. 366 (2016), no. 3-4, 1281-1317

    work page 2016

  44. [44]

    Yau, Harmonic functions on complete Riemannian manifolds, Comm

    S.-T. Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math., 28 (1975), 201-228

    work page 1975

  45. [45]

    Yau, Nonlinear analysis in geometry, L'Enseign

    S.-T. Yau, Nonlinear analysis in geometry, L'Enseign. Math., 33 (1987), 109-158

    work page 1987