Pith. sign in

REVIEW 1 major objections 3 minor 57 references

On a compact Einstein four-manifold, pointwise simplicity of the largest eigenvalue of the self-dual Weyl tensor forces the metric to be conformal to a Kähler metric with positive scalar curvature, up to a double cover.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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

For compact Einstein four-manifolds, the spectral gap α>β in the self-dual Weyl curvature forces β=γ=−α/2, so the metric is conformally Kähler with positive scalar curvature (after at most a double cover).

T0 review reviewed 2026-08-01 challenge →

load-bearing objection The compact ρ<1 rigidity theorem is strong, the algebra holds up, and the sharpness examples are convincing; the noncompact Theorem 1.9 has an unjustified zero-set step that needs a fix before I'd trust its conclusion. the 1 major comments →

arxiv 2607.21538 v1 pith:6VH2V5HT submitted 2026-07-23 math.DG

Conformal K\"{a}hler rigidity of Einstein four-manifolds

Giovanni Catino , Davide Dameno This is my paper
classification math.DG MSC 53C2553C2453C1853C2153C55
keywords Einstein four-manifoldsharmonic self-dual Weyl curvatureconformally Kähler metricsgravitational instantonsKähler–Einstein surfaceseigenvalue simplicityzero-capacity argumentholomorphic sectional curvature
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The reading

This paper proves a rigidity statement for four-dimensional Einstein manifolds, and more generally for metrics whose self-dual Weyl curvature is harmonic. The central claim is that if the largest eigenvalue of the self-dual Weyl tensor is everywhere strictly larger than the second one—equivalently, the ratio ρ = β/α is strictly below 1—then, after at worst passing to a double cover, the metric is conformal to a Kähler metric with positive scalar curvature. In fact, the ratio must jump to its lowest possible value, ρ = −1/2, which is the signature of a conformal Kähler structure. For Einstein metrics, the same conclusion holds under a uniform gap β ≤ ρ0 α with ρ0 < 1, even across points where W^+ vanishes; adapting the argument to complete Ricci-flat four-manifolds yields new constraints on gravitational instantons. The paper also turns the gap condition into a sharp pinching theorem for the holomorphic sectional curvature of compact Kähler–Einstein surfaces, and constructs examples showing the hypotheses cannot be relaxed.

Core claim

The paper establishes that a pointwise simplicity condition on the largest eigenvalue of the self-dual Weyl tensor—α_h > β_h, equivalently the conformally invariant ratio ρ = β_h/α_h < 1—forces ρ = −1/2 identically on a compact manifold with harmonic self-dual Weyl curvature. Then the conformally related metric g = α_h^{2/3} h is Kähler with positive scalar curvature. For Einstein metrics, a uniform gap β_h ≤ ρ0 α_h with ρ0 < 1 is enough even if W^+ vanishes somewhere: either W^+ ≡ 0, or W^+ is nowhere zero and the same conclusion holds. The noncompact analogue covers complete Ricci-flat four-manifolds satisfying volume-growth and curvature-decay bounds, and a separate theorem gives a sharp

What carries the argument

The central object is the self-dual Weyl operator W^+ and its eigenvalues α ≥ β ≥ γ. The argument runs through the conformal change g = α_h^{2/3} h, which makes the largest eigenvalue of the weighted tensor W^+ = α_h^{−1/3} W^+_g constantly 1, so the ratio ρ = β/α becomes the only spectral parameter. A system of new first-order identities relates the connection 1-forms a and c of the eigenframe by the ratio k = (1−ρ)/(2+ρ); substituting these into a weighted Weitzenböck formula yields an exact pointwise identity in which the indefinite gradient term is pinned down. An integral of this identity has a nonnegative integrand that vanishes only when ∇ω₁ = 0 and ρ = −1/2. A zero-capacity cutoff ar

Load-bearing premise

The proof that rigidity survives the points where W^+ vanishes rests entirely on the zero set {W^+ = 0} being small enough—countably 2-rectifiable with finite H²-measure, hence codimension at least two—so that the cutoff functions used in the capacity argument have vanishing gradient energy; if that zero set were larger, the argument would only yield conformal Kähler rigidity on the complement.

What would settle it

Find a compact oriented Einstein four-manifold with W^+ not identically zero and β ≤ ρ0 α for some ρ0 < 1 whose metric is not conformal to a Kähler metric with positive scalar curvature; a more targeted check is to exhibit an Einstein metric on which the zero set {W^+ = 0} has positive 2-dimensional Hausdorff content, which would directly violate the cutoff estimate (4.11) and invalidate Theorem 1.4.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Classification corollary: every compact simply connected Einstein four-manifold with positive scalar curvature and a uniformly simple largest eigenvalue is isometric to the round S⁴ or CP², a Kähler–Einstein del Pezzo surface, or one of the two known non-Kähler Hermitian Einstein metrics (Page or Chen–LeBrun–Weber).
  • No uniform gap for nonpositive scalar curvature: a compact Einstein four-manifold with s_h ≤ 0 and W^+ not identically zero must have sup ρ = 1 on the set where W^+ ≠ 0.
  • Kähler–Einstein pinching: if a compact Kähler–Einstein surface with s_h ≤ 0 satisfies Θ < 2/3 between the average and minimum holomorphic sectional curvature, then W^- ≡ 0 and the surface is either flat up to finite cover or a compact ball quotient; the bound 2/3 is optimal in the sense that negative-curvature examples exist.
  • Ricci-flat rigidity: complete Ricci-flat four-manifolds satisfying β ≤ ρ0 α with ρ0 < 1 and the stated volume/curvature bounds are conformally Kähler with positive scalar curvature and hence Hermitian non-Kähler; in the gravitational-instanton setting this yields a type-II classification (ALE for Euclidean volume growth, Kerr/Chen–Teo/Taub-bolt/reversed Taub–NUT for cubic growth).
  • Sharpness: K3 surfaces with Calabi–Yau metrics and reversed orientation, and multicentered Gibbons–Hawking instantons, have points with α = β > 0, so the uniform gap cannot be relaxed without losing the conclusion.

Where Pith is reading between the lines

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

  • Stability estimate: the method's exact algebraic evaluation of the gradient term suggests a quantitative converse—if ρ is bounded above by ρ0 < 1, then the distance (in some conformal norm) between the conformal metric and the resulting Kähler metric might be controlled by ρ0 + 1/2; the paper does not state such an estimate.
  • Transferability of the capacity argument: the same template—divergence-free curvature-type tensor, weighted Weitzenböck formula, and elliptic zero-set control—applies to other geometric first-order systems, such as harmonic self-dual 2-forms or Dirac-type equations on four-manifolds.
  • Boundary phenomenon at ρ = 1: the Gibbons–Hawking example shows ρ has a jump discontinuity through the zero set of W^+, suggesting that the locus where α = β > 0 functions as a phase boundary; understanding this locus could connect to the topology of the maximal-eigenvalue line bundle.
  • Noncompact sharpening: Theorem 6.3 is stated under cubic volume growth, but the proof uses only Hölder's inequality and L² curvature, so the same zero-capacity technique might extend the classification to other collapsed ends such as ALG- or ALH-type gravitational instantons.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 3 minor

Summary. The paper proves rigidity results for oriented Riemannian four-manifolds whose self-dual Weyl curvature is harmonic. The main compact statement is Theorem 1.2: if δ_h W_h^+ = 0 and the largest eigenvalue α_h of W_h^+ is everywhere strictly larger than the second eigenvalue β_h, then, after at worst passing to a double cover, the metric h is conformal to a Kähler metric with positive scalar curvature; equivalently, the ratio ρ = β_h/α_h is forced to be -1/2. For compact Einstein metrics with W_h^+ not identically zero, Theorem 1.4 proves the same conclusion under the uniform gap β_h ≤ ρ_0 α_h with ρ_0 < 1, using a zero-capacity argument across the zero set of W_h^+. The paper also gives a pinching theorem for compact Kähler-Einstein surfaces of nonpositive scalar curvature (Theorem 1.8), a noncompact Ricci-flat extension (Theorem 1.9), classification consequences for gravitational instantons, and sharpness examples from K3 surfaces and multicentered Gibbons-Hawking instantons. The proof centers on LeBrun's conformal normalization g = α_h^{2/3} h, new first-order identities for the eigenvalues of W_g^+ on its regular spectral set, and the integral identity (3.9) whose integrand is a sum of two nonnegative terms.

Significance. If the compact results hold, they are a substantial improvement over previous work of Wu and LeBrun: the strict simplicity condition α_h > β_h alone forces ρ = -1/2, with no extra pinching constant. The algebraic chain leading to (3.8)-(3.9) is explicit, parameter-free, and internally consistent; the sharpness examples in Section 7 are valuable. The zero-capacity method is a promising tool for crossing the zero set of W_h^+ and for noncompact problems. The paper does not rely on fitted constants or circular predictions; its main limitation is a gap in the noncompact zero-set step described below.

major comments (1)
  1. [Section 6, Theorem 1.9; also Section 4, Theorem 1.4] After Proposition 4.1 yields β_h = γ_h = -α_h/2 on X = M\N, the proof concludes N = ∅ by invoking Derdziński [16, Proposition 5 (iv)] without stating its hypotheses. The cited proposition is presented in the introduction as a result on zeros of W_h^+ for Einstein four-manifolds, and in Derdziński's paper the relevant setting is compact. The manuscript does not verify that Proposition 5(iv) applies to a complete noncompact Ricci-flat manifold. This step is load-bearing: the capacity estimate (4.11)/(6.1) controls only the energy of the cutoff functions and proves rigidity on X; it does not by itself show that the parallel Kähler form, or the double cover, extends across N. Unless the authors state Proposition 5(iv) and verify its hypotheses in the noncompact setting, Theorem 1.9 and the dependent Corollary 6.1 and Theorem 6.3 are established only on M\N, not on all of M.
minor comments (3)
  1. [Section 4, Proposition 4.1] The sentence 'If the line subbundle L associated to the eigenspace of α_h is not orientable, We recall that ...' appears garbled and should be rewritten.
  2. [Section 3, Eq. (3.6)] The factor 2 in W^+(∇_e ω_1,∇_e ω_1) = 2(β|a|^2 + γ|c|^2) is implicit in the normalization |ω_i|^2 = 2; a brief remark would help the reader.
  3. [Section 4, Eq. (4.8)] The derivation of the uniform bound ∑ r_i^2 ≤ C_N from finite Hausdorff measure would benefit from a one-line explanation or a more precise reference to the δ-content comparison.

Circularity Check

0 steps flagged

No significant circularity: the rigidity conclusion follows from parameter-free, first-order Weitzenböck identities and the spectral gap; the only caveat is an external-import hypothesis check in Theorem 1.9, not a circular reduction.

full rationale

I traced the derivation chain from the spectral hypothesis to the rigidity conclusion. The core step is the weighted Weitzenböck/integral identity (3.5), combined with the new first-order identities (2.4)-(2.5) and the evaluation P = Q_rho |\nabla \omega_1|^2 in (3.7)-(3.8). This is a genuine identity: no fitted constants, no parameter chosen from the data, and no 'prediction' that is equivalent to the assumption by construction. The conclusion rho = -1/2 is obtained by summing two nonnegative terms in (3.9), which is a legitimate analytic consequence of the divergence-free condition and the spectral gap rho < 1. The proof relies on standard external tools (LeBrun's conformal normalization, Derdziński's Weitzenböck formula, Bär's zero-set theorem), but these are cited as established results with independent content, not as self-citations by the present authors. The paper contains essentially no self-citation load-bearing argument: the only self-reference is a standard reference to [11] for conformal covariance, which is not load-bearing. The reader's concern about Theorem 1.9 is a hypothesis-checking issue, not circularity: the proof says 'we can use our assumption and [16, Proposition 5 (iv)] to deduce that N = ∅', and the paper does not state the hypotheses of [16, Proposition 5(iv)] in the noncompact setting. If that proposition requires compactness, Theorem 1.9 would have a correctness gap, but this is not a circular step because the proposition is an external result, not an assumption built into the claimed conclusion. Thus the appropriate finding is no significant circularity, score 0.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

The central theorems are derived from the Einstein/divergence structure rather than postulated; the ledger records that the main external inputs are unique-continuation/elliptic-system results (Bär, Polombo, Derdziński) and the Kähler/Einstein classification tools used in examples. There are no free parameters because ρ0, θ and c are hypotheses, not fitted values.

axioms (6)
  • domain assumption Bär's zero-set theorem [4, Cor 1]: the zero set of a non-trivial solution of a first-order elliptic system is countably 2-rectifiable with finite H^2 measure (codimension at least 2).
    Used in Theorem 1.4 and Theorem 1.9 to justify the cutoff construction; the estimate (4.11) fails if the zero set is larger.
  • domain assumption On an Einstein four-manifold, W^+ satisfies the first-order elliptic system D W^+=0 (Bianchi + divergence-free), cited to [49].
    Precondition for applying Bär's theorem in Section 4.
  • domain assumption Derdziński's zero-set rigidity [16, Prop 5(iv)]: with #spec(W^+)≤2 everywhere and W^+ not identically zero, W^+ never vanishes.
    Load-bearing in Theorems 1.4 and 1.9 to conclude N=∅ and extend Kähler rigidity from X to M.
  • domain assumption Derdziński's Weitzenböck formula and Prop 5 [16]: the weighted Weitzenböck identity (2.9) and the characterization of g Kähler (∇ω1=0, eigenvalues s/6, −s/12, −s/12).
    Used in Lemma 2.5, Theorem 1.2, and the converse direction.
  • standard math LeBrun's conformal normalization [36]: the α-eigenspace line bundle is trivial after the double cover, and the conformal covariance of the divergence (Lemma 2.1, proved in the paper).
    Core setup of Section 2.
  • domain assumption Lye's theorem [42, Thm 4.6] and Artebani–Sarti [2] on K3 automorphisms of order 3 fixing a genus >1 curve; Yau's Calabi conjecture [57] for Ricci-flat Kähler metrics.
    Used only for the sharpness examples in Section 7, not for the main rigidity theorems.

reviewed 2026-08-01 · how reviews work

0 comments
Cite this review

Pith. "Pith review of Conformal K\"{a}hler rigidity of Einstein four-manifolds." pith.science (2026). https://pith.science/paper/6VH2V5HT

@misc{pith2026260721538,
  author       = {Pith},
  title        = {Pith review of: Conformal K\"ahler rigidity of Einstein four-manifolds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6VH2V5HT}},
  note         = {Machine review of arXiv:2607.21538}
}
Share X Bluesky LinkedIn Reddit HN
abstract

For a compact, connected, oriented Einstein four-manifold, we prove that, if the largest eigenvalue of the self-dual Weyl curvature $W^+$ is everywhere simple, then, after at worst passing to a double cover, the metric is conformally K\"ahler with positive scalar curvature; more generally, this result holds for metrics with harmonic self-dual Weyl curvature. For Einstein metrics satisfying a uniform simplicity hypothesis on the largest eigenvalue, we further prove that either $W^+\equiv 0$, or $W^+$ nowhere vanishes and the previous conclusion holds. We also obtain extensions to complete Ricci-flat four-manifolds and an optimal pinching theorem for the holomorphic sectional curvature of compact K\"ahler--Einstein surfaces. The proof combines LeBrun's conformal normalization with the resulting weighted divergence equation and new first-order identities. A zero-capacity argument allows this method to be used across the zero set of $W^+$ and at infinity in the noncompact case. Finally, K3 surfaces and multicentered Gibbons--Hawking gravitational instantons show that our assumptions are sharp.

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

57 extracted references · 4 linked inside Pith

  1. [1]

    M. T. Anderson, P. B. Kronheimer, C. LeBrun,Complete Ricci-flat Kähler manifolds of infinite topological type, Comm. Math. Phys.125(1989), no. 4, 637–642

    1989

  2. [2]

    Artebani, A

    M. Artebani, A. Sarti,Non-symplectic automorphisms of order 3 on K3 surfaces, Math. Ann.342(2008), no. 4, 903–921

    2008

  3. [3]

    Bando, A

    S. Bando, A. Kasue, H. Nakajima,On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth, Invent. Math.97 (1989), no. 2, 313–349

    1989

  4. [4]

    Bär,Zero sets of solutions to semilinear elliptic systems of first order, Invent

    C. Bär,Zero sets of solutions to semilinear elliptic systems of first order, Invent. Math. 138(1999), no. 1, 183–202

    1999

  5. [5]

    Berger, Sur les variétés d’Einstein compactes, inComptes Rendus de la IIIe Réunion du Groupement des Mathématiciens d’Expression Latine (Namur, 1965), pp

    M. Berger, Sur les variétés d’Einstein compactes, inComptes Rendus de la IIIe Réunion du Groupement des Mathématiciens d’Expression Latine (Namur, 1965), pp. 35–55, Librairie Universitaire, Louvain

    1965

  6. [6]

    A. L. Besse,Einstein manifolds, reprint of the 1987 edition, Classics in Mathematics, Springer, Berlin, 2008. 37

    1987

  7. [7]

    Biquard, P

    O. Biquard, P. Gauduchon,On toric Hermitian ALF gravitational instantons, Comm. Math. Phys.399(2023), no. 1, 389–422

    2023

  8. [8]

    Biquard, P

    O. Biquard, P. Gauduchon, and C. LeBrun,Gravitational instantons, Weyl curvature, and conformally Kähler geometry, Int. Math. Res. Not. IMRN2024, no. 20, 13295– 13311

  9. [9]

    Bochner,Curvature and Betti numbers

    S. Bochner,Curvature and Betti numbers. II, Ann. of Math. (2)50(1949), 77–93

    1949

  10. [10]

    Bourguignon, Les variétés de dimension4à signature non nulle dont la courbure est harmonique sont d’Einstein, Invent

    J.-P. Bourguignon, Les variétés de dimension4à signature non nulle dont la courbure est harmonique sont d’Einstein, Invent. Math.63(1981), no. 2, 263–286

    1981

  11. [11]

    Catino, P

    G. Catino, P. Mastrolia,A perspective on canonical Riemannian metrics, Progress in Mathematics, vol. 336, Birkhäuser/Springer, Cham, 2020

    2020

  12. [12]

    Cheeger, G

    J. Cheeger, G. Tian,Curvature and injectivity radius estimates for Einstein 4- manifolds, J. Amer. Math. Soc.19(2006), no. 2, 487–525

    2006

  13. [13]

    G. Chen, X. Chen,Gravitational instantons with faster than quadratic curvature decay (I), Acta Math.227(2021), no. 2, 263–307

    2021

  14. [14]

    X. X. Chen, C. LeBrun, B. Weber,On conformally Kähler, Einstein manifolds, J. Amer. Math. Soc.21(2008), no. 4, 1137–1168

    2008

  15. [15]

    Y. Chen, E. Teo,A new AF gravitational instanton, Phys. Lett. B703(2011), no. 3, 359–362

    2011

  16. [16]

    Derdziński,Self-dual Kähler manifolds and Einstein manifolds of dimension four, Compositio Math.49(1983), no

    A. Derdziński,Self-dual Kähler manifolds and Einstein manifolds of dimension four, Compositio Math.49(1983), no. 3, 405–433

    1983

  17. [17]

    Dunajski,Gravitational instantons, old and new, Acta Phys

    M. Dunajski,Gravitational instantons, old and new, Acta Phys. Polon. B55(2024), no. 12, Paper No. A3, 18 pp

    2024

  18. [18]

    Eguchi, A

    T. Eguchi, A. J. Hanson,Self-dual solutions to Euclidean gravity, Ann. Physics120 (1979), no. 1, 82–106

    1979

  19. [19]

    L. C. Evans, R. F. Gariepy,Measure theory and fine properties of functions, revised ed., Textbooks in Mathematics, CRC Press, Boca Raton, FL, 2015

    2015

  20. [20]

    Friedrich, H

    T. Friedrich, H. Kurke,Compact four-dimensional self-dual Einstein manifolds with positive scalar curvature, Math. Nachr.106(1982), 271–299

    1982

  21. [21]

    G. W. Gibbons, S. W. Hawking,Action integrals and partition functions in quantum gravity, Phys. Rev. D15(1977), 2752–2756

    1977

  22. [22]

    G. W. Gibbons, S. W. Hawking,Gravitational multi-instantons, Phys. Lett. B78 (1978), no. 4, 430–432

    1978

  23. [23]

    G. W. Gibbons, S. W. Hawking,Classification of gravitational instanton symmetries, Comm. Math. Phys.66(1979), no. 3, 291–310

    1979

  24. [24]

    3, 475–546

    M.Gross, P.M.H.Wilson,Large complex structure limits ofK3surfaces, J.Differential Geom.55(2000), no. 3, 475–546

    2000

  25. [25]

    Guan,On bisectional nonpositively curved compact Kähler–Einstein surfaces, Pacific J

    D. Guan,On bisectional nonpositively curved compact Kähler–Einstein surfaces, Pacific J. Math.288(2017), no. 2, 343–353

    2017

  26. [26]

    Guenancia, U

    H. Guenancia, U. Hamenstädt,Kähler–Einstein metrics of negative curvature, preprint, arXiv:2503.02838v3, 2026

    Pith/arXiv arXiv 2026

  27. [27]

    N. J. Hitchin,Compact four-dimensional Einstein manifolds, J. Differential Geom.9 (1974), 435–441

    1974

  28. [28]

    N. J. Hitchin,Kählerian twistor spaces, Proc. London Math. Soc. (3)43(1981), no. 1, 133–150

    1981

  29. [29]

    Kobayashi,Fixed points of isometries, Nagoya Math

    S. Kobayashi,Fixed points of isometries, Nagoya Math. J.13(1958), 63–68

    1958

  30. [30]

    Kobayashi, K

    S. Kobayashi, K. Nomizu,Foundations of differential geometry. Vol. II, reprint of the 1969 original, Wiley Classics Library, Wiley-Interscience, Wiley, New York, 1996

    1969

  31. [31]

    P. B. Kronheimer,The construction of ALE spaces as hyper-Kähler quotients, J. Differential Geom.29(1989), no. 3, 665–683

    1989

  32. [32]

    P. B. Kronheimer,A Torelli-type theorem for gravitational instantons, J. Differential Geom.29(1989), no. 3, 685–697

    1989

  33. [33]

    LeBrun,On Einstein, Hermitian 4-manifolds, J

    C. LeBrun,On Einstein, Hermitian 4-manifolds, J. Differential Geom.90(2012), no. 2, 277–302. 38

    2012

  34. [34]

    LeBrun,Einstein manifolds and extremal Kähler metrics, J

    C. LeBrun,Einstein manifolds and extremal Kähler metrics, J. Reine Angew. Math. 678(2013), 69–94

    2013

  35. [35]

    LeBrun,Einstein metrics, harmonic forms, and symplectic four-manifolds, Ann

    C. LeBrun,Einstein metrics, harmonic forms, and symplectic four-manifolds, Ann. Global Anal. Geom.48(2015), no. 1, 75–85

    2015

  36. [36]

    LeBrun,Einstein manifolds, self-dual Weyl curvature, and conformally Kähler geometry, Math

    C. LeBrun,Einstein manifolds, self-dual Weyl curvature, and conformally Kähler geometry, Math. Res. Lett.28(2021), no. 1, 127–144

    2021

  37. [37]

    Li,On 4-dimensional Ricci-flat ALE manifolds, preprint, arXiv:2304.01609, 2023

    M. Li,On 4-dimensional Ricci-flat ALE manifolds, preprint, arXiv:2304.01609, 2023

    Pith/arXiv arXiv 2023

  38. [38]

    Li,Classification results for conformally Kähler gravitational instantons, preprint, arXiv:2310.13197, 2023

    M. Li,Classification results for conformally Kähler gravitational instantons, preprint, arXiv:2310.13197, 2023

    Pith/arXiv arXiv 2023

  39. [39]

    M. Li, S. Sun,Geometry of gravitational instantons, to appear inSurveys in Differential Geometry, arXiv:2606.10443, 2026

    Pith/arXiv arXiv 2026

  40. [40]

    Lichnerowicz,Spineurs harmoniques, C

    A. Lichnerowicz,Spineurs harmoniques, C. R. Acad. Sci. Paris257(1963), 7–9

    1963

  41. [41]

    J. D. Lotay, G. Oliveira,Special Lagrangians, Lagrangian mean curvature flow and the Gibbons-Hawking ansatz, J. Differential Geom.126(2024), no. 3, 1121–1184

    2024

  42. [42]

    J. O. Lye,Geodesics on a K3 surface near the orbifold limit, Ann. Global Anal. Geom. 63(2023), Paper No. 20

    2023

  43. [43]

    Matsumoto, S

    M. Matsumoto, S. Tanno,Kählerian spaces with parallel or vanishing Bochner curva- ture tensor, Tensor (N.S.)27(1973), 291–294

    1973

  44. [44]

    Minerbe,On the asymptotic geometry of gravitational instantons, Ann

    V. Minerbe,On the asymptotic geometry of gravitational instantons, Ann. Sci. Ècole Norm. Sup. (4)43(2010), no. 6, 883–924

    2010

  45. [45]

    Odaka, C

    Y. Odaka, C. Spotti, S. Sun,Compact moduli spaces of del Pezzo surfaces and Kähler–Einstein metrics, J. Differential Geom.102(2016), no. 1, 127–172

    2016

  46. [46]

    D. N. Page,Taub-NUT instanton with an horizon, Phys. Lett. B78(1978), no. 2, 249–251

    1978

  47. [47]

    D. N. Page,A compact rotating gravitational instanton, Phys. Lett. B79(1979), 235–238

    1979

  48. [48]

    Penrose, W

    R. Penrose, W. Rindler,Spinors and space-time. Vol. 2, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1986

    1986

  49. [49]

    Polombo,De nouvelles formules de Weitzenböck pour des endomorphismes har- moniques

    A. Polombo,De nouvelles formules de Weitzenböck pour des endomorphismes har- moniques. Applications géométriques, Ann. Sci. Ècole Norm. Sup. (4)25(1992), no. 4, 393–428

    1992

  50. [50]

    J. Ross, R. P. Thomas,Weighted projective embeddings, stability of orbifolds, and constant scalar curvature Kähler metrics, J. Differential Geom.88(2011), no. 1, 109–159

    2011

  51. [51]

    Y.-T. Siu, P. Yang,Compact Kähler–Einstein surfaces of nonpositive bisectional curvature, Invent. Math.64(1981), no. 3, 471–487

    1981

  52. [52]

    S. Sun, R. Zhang,Collapsing geometry of hyperkähler four-manifolds and applications, Acta Math.232(2024), 325–424

    2024

  53. [53]

    Tachibana,On the Bochner curvature tensor, Natur

    S. Tachibana,On the Bochner curvature tensor, Natur. Sci. Rep. Ochanomizu Univ. 18(1967), 15–19

    1967

  54. [54]

    Tian,On Calabi’s conjecture for complex surfaces with positive first Chern class, Invent

    G. Tian,On Calabi’s conjecture for complex surfaces with positive first Chern class, Invent. Math.101(1990), no. 1, 101–172

    1990

  55. [55]

    Tricerri, L

    F. Tricerri, L. Vanhecke,Curvature tensors on almost Hermitian manifolds, Trans. Amer. Math. Soc.267(1981), no. 2, 365–397

    1981

  56. [56]

    Wu,Einstein four-manifolds with self-dual Weyl curvature of nonnegative determi- nant, Int

    P. Wu,Einstein four-manifolds with self-dual Weyl curvature of nonnegative determi- nant, Int. Math. Res. Not. IMRN2021, no. 2, 1043–1054

  57. [57]

    Giuseppe Peano

    S.-T. Yau,On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation. I, Comm. Pure Appl. Math.31(1978), no. 3, 339–411. 39 Dipartimento di Matematica, Politecnico di Milano, Italy Email address:giovanni.catino@polimi.it Dipartimento di Matematica “Giuseppe Peano”, Università degli Studi di Torino, Italy Email address:davide...

    1978

This paper was first reviewed by deepseek-v4-flash on August 1, 2026.