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On any compact toric contact manifold of Reeb type, there are T-invariant CR structures whose CR Yamabe invariants take negative values, and when the manifold is transversally Fano they take all three signs.

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2026-07-31 00:47 UTC pith:KPQEWB2T

load-bearing objection Solid toric reduction that answers the sign question for CR Yamabe invariants; main caveat is a mild higher-dim hypothesis that is fixable. the 2 major comments →

arxiv 2607.27441 v1 pith:KPQEWB2T submitted 2026-07-29 math.DG

The toric CR Yamabe problem

Eveline Legendre , Carlo Scarpa , Christina W. T{\o}nnesen-Friedman This is my paper
classification math.DG MSC 53C2558E1153C1832Q1553D1032V20
keywords CR Yamabe problemtoric Sasaki manifoldsTanaka-Webster scalar curvatureEinstein-Hilbert functionalsymplectic potentialsmoment polytopeequivariant Yamabe invariant
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 paper studies the torus-equivariant CR Yamabe problem on compact co-oriented contact manifolds that carry a fixed-point-free torus action of Reeb type. In the toric case the problem reduces to a boundary-value elliptic PDE for a pair of functions on the moment polytope: a normalised symplectic potential that encodes the CR structure and a positive conformal factor. By evaluating a reduced Einstein–Hilbert energy along explicit paths of potentials, the authors prove that the equivariant CR Yamabe energy can be driven to minus infinity or to non-positive values. Consequently every such manifold admits a T-invariant CR structure with negative Yamabe invariant, and every transversally Fano example admits structures with negative, zero and positive invariants. The same reduction supplies many explicit constant Tanaka–Webster curvature examples and shows that constant-scalar-curvature Sasaki structures sit at the boundary of the convex hull of constant-curvature conformal structures.

Core claim

For any compact toric contact manifold of Reeb type the T-equivariant CR Yamabe problem is equivalent to an elliptic boundary-value problem on the moment polytope. Along paths of symplectic potentials that become singular or go to infinity, the reduced Einstein–Hilbert energy tends to non-positive values or to minus infinity, so the manifold always carries T-invariant CR structures of negative CR Yamabe invariant; when it is transversally Fano it also carries structures of zero and positive invariant.

What carries the argument

The reduced CR Einstein–Hilbert functional EH(φ,f) = (2∫_∂P f^{-n} dσ + n∫_P f^{-n-1} ⟨H_φ, Hess f⟩ dx) / (∫_P f^{-n-1} dx)^{n/(n+1)}, defined on pairs of a normalised symplectic potential φ and a positive function f on the moment polytope P; its critical points are precisely the T-invariant constant Tanaka–Webster structures, and its limiting behaviour along paths of potentials controls the sign of the equivariant Yamabe energy.

Load-bearing premise

In higher dimensions the energy is proved to go to minus infinity only when the Hessian of the limiting potential vanishes to order strictly higher than the dimension of the polytope along a hyperplane through an interior point; without that quantitative vanishing the integral against the test function need not diverge.

What would settle it

Construct an explicit path of symplectic potentials on a polytope of dimension greater than one whose Hessian vanishes only to order n−1 or lower at an interior degeneracy, and check whether the reduced energy still tends to minus infinity; if it remains bounded the case-2 claim fails.

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

If this is right

  • Every compact toric contact manifold of Reeb type admits a T-invariant CR structure with negative CR Yamabe invariant.
  • Every transversally Fano toric contact manifold admits T-invariant CR structures realising all three signs of the CR Yamabe invariant.
  • Constant-scalar-curvature Sasaki structures cannot lie in the interior of the convex hull of constant Tanaka–Webster structures in the same conformal class.
  • The equivariant Yamabe energy is controlled by a simpler functional I(f) on concave positive functions on the polytope, linking it to K-stability of the toric Sasaki manifold.
  • Explicit non-Sasaki constant Tanaka–Webster examples exist that are not absolute minimisers of the Einstein–Hilbert energy in their T-invariant conformal class.

Where Pith is reading between the lines

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

  • The same polytope reduction should let one decide whether a constant-scalar-curvature Sasaki metric that realises the global minimum of the Einstein–Hilbert energy on the Reeb cone is automatically a T-invariant CR Yamabe minimiser in its conformal class.
  • Extending the energy continuously to weakly convex or piecewise-linear potentials would turn the question of the sign of the toric CR Yamabe invariant into a purely convex-geometric problem on the moment polytope.
  • The appearance of both signs on a fixed contact structure suggests that the ordinary (non-equivariant) CR Yamabe invariant may also change sign under deformation of the CR structure, even without torus symmetry.

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

2 major / 5 minor

Summary. The paper studies the T-equivariant CR Yamabe problem on compact co-oriented contact manifolds of Reeb type carrying a fixed-point-free torus action of maximal dimension. Using the Guillemin–Abreu–Martelli–Sparks–Yau dictionary, the authors reduce T-invariant constant Tanaka–Webster scalar curvature to a nonlinear elliptic boundary-value problem (Prop. 1.2 / Eq. (2)) for a pair (symplectic potential φ, conformal factor f) on the labelled moment polytope P. They derive an explicit formula for the reduced Einstein–Hilbert functional EH(φ,f) and prove (Thm. 1.4) that along paths of normalised symplectic potentials that either escape to infinity with strictly convex leading term or hit a controlled degeneracy of the Hessian, the equivariant CR Yamabe energy Y^T_CR tends to a non-positive value or to −∞ respectively. Corollaries assert that every such contact toric manifold admits a T-invariant CR structure of negative Yamabe invariant, and that transversally Fano ones realise all three signs. Further results include a convex-hull obstruction for cscS structures (Prop. 1.8), an extension of EH to piecewise-linear geodesic ribbons via an action functional (Prop. 1.7 / §5), and a collection of explicit cscTW and sign-changing examples (§6).

Significance. The work supplies the first systematic dimensional reduction of the equivariant CR Yamabe problem in the toric setting and gives concrete, computable control on the sign of Y^T_CR. The answer to the question of Sung–Takeuchi on the existence of compatible CR structures of both positive and negative Yamabe invariant on a fixed contact structure is of clear interest. The reduced functional I(f) and its link to toric K-stability (Question 5.3) open a natural bridge between CR Yamabe theory and the existing Donaldson–Futaki / weighted cscK literature. The explicit non-Sasaki cscTW examples on the trivial Hirzebruch surface and the sign-changing deformations in §§6.1–6.3 are useful concrete data. The core analytic arguments (test-function estimates with piecewise-linear creases and radial powers) are standard and transparent once the toric dictionary is in place.

major comments (2)
  1. [Theorem 1.4 (Case 2), Corollary 1.5, Lemma 4.1 / Corollary 4.2] Corollary 1.5 asserts existence of a T-invariant CR structure with Y^T_CR < 0 in every dimension, yet Case 2 of Theorem 1.4 requires the quantitative vanishing-order hypothesis α > n−1 on ⟨v_T, (Hess φ_T) v_T⟩ along the crease (Lemma 4.1 / Cor. 4.2). The paper only remarks that the condition is automatic when dim N = 3. An explicit construction (or a short existence argument) of paths in S_o(P) that realise this order in all dimensions is needed for the corollary to hold as stated; without it the higher-dimensional claim rests on an unverified hypothesis.
  2. [Corollary 1.6, §1.1] Corollary 1.6 claims the existence of a T-invariant CR structure with Y^T_CR = 0 on every transversally Fano toric contact manifold. Positive sign follows from the Sasaki–Einstein metric (FOW09) together with CR-Obata uniqueness; negative sign follows from Cor. 1.5. The zero value is not constructed and is not justified by an intermediate-value argument: the paper does not establish continuity of the map φ ↦ Y^T_CR(φ) on S_o(P) (or on any path connecting a positive structure to a negative one). Either a continuity statement or an explicit zero-energy example should be supplied.
minor comments (5)
  1. [Proposition 1.2] Proposition 1.2: the phrase “the value of the of Tanaka-Webster scalar curvature” contains a duplicated article; also the constant C should be identified with the value of EH up to the usual positive normalisation factor already used later in the text.
  2. [Figure 1 (§4)] Figure 1 is a helpful schematic but the regions labelled Y_CR ≤ 0 / ≥ 0 / < 0 are not rigorously delimited by the theorems; a caption clarifying that the figure is only heuristic would avoid over-interpretation.
  3. [Throughout] Several typographical artefacts appear in the extracted text (“T oric”, “Y amabe”, “W e will use”, stray spaces in section headings). A careful proof-reading pass is recommended before final submission.
  4. [§5, equations (16)–(17)] In §5 the passage from the smooth action functional A^χ_s(t) to the singular Monge–Ampère expression (17) invokes Rudin’s theorems on Lebesgue decomposition; a one-sentence reminder that the singular part of MA(φ_∞) is supported on the crease set (hence has vanishing Radon–Nikodym derivative) would make the justification self-contained.
  5. [Question 1.9, §6.2.3] Question 1.9 is well-motivated by the examples of §6.2 and §6.4; it would help the reader if the authors briefly indicated whether any of the explicit non-Sasaki cscTW solutions constructed in §6.2.3 could serve as counter-examples once the EH = EH_min hypothesis is dropped (they already show they are not absolute minimisers).

Circularity Check

0 steps flagged

No significant circularity: sign limits of the toric CR Yamabe energy are direct test-function estimates on a reduced Einstein–Hilbert functional.

full rationale

The load-bearing claims (Theorem 1.4 and Corollaries 1.5–1.6) are obtained by writing the T-invariant Einstein–Hilbert functional in action-angle coordinates via the Abreu formula and the standard integration-by-parts identity on the Delzant polytope, then evaluating it on explicit test functions (piecewise-linear creases for degenerating paths; radial powers for paths to infinity). The resulting limits ≤0 and −∞ are elementary integral estimates; they do not encode the conclusion in a normalization, fit, or uniqueness theorem. Positive Yamabe sign for transversally Fano examples uses the external Futaki–Ono–Wang existence of Sasaki–Einstein metrics, not a self-referential uniqueness import. Self-citations (LLS23/25, BHLTF, Leg11, ACMY24) supply background dictionary and variational characterizations whose statements are independent of the new sign theorems. No step reduces a claimed prediction to a fitted input or to a definition that already contains the result.

Axiom & Free-Parameter Ledger

0 free parameters · 7 axioms · 2 invented entities

The paper is pure mathematics. Load-bearing input is standard CR/Sasaki/toric geometry plus a few analytic lemmas; no empirical free parameters. The only extra structural hypothesis that is not automatic is the vanishing-order condition in higher-dimensional case 2 of Theorem 1.4.

axioms (7)
  • domain assumption Classification of compact contact toric manifolds of Reeb type by good strictly convex moment cones (Lerman, BM93, BG00).
    Used throughout §2 to identify (N,D,T) with a labeled polytope P_ξ.
  • domain assumption Guillemin–Abreu–Martelli–Sparks–Yau correspondence: toric Sasaki structures ↔ symplectic potentials with prescribed boundary singularities (Prop. 2.3–2.4).
    Defines the space S(P) and the matrix H = (Hess φ)^{-1} on which all curvature formulas rest.
  • domain assumption Abreu formula for transversal/Tanaka–Webster scalar curvature Scal_TW = −(H^{ij})_{,ij} and its conformal transformation law (6).
    Starting point for the cscTW PDE (7) and the reduced EH functional (9).
  • standard math Integration-by-parts identity ∫_P ψ_1 (H^{ij} ψ_0)_{ij} dx = −2∫_∂P ψ_1 ψ_0 dσ + ∫_P ψ_0 ⟨H,Hess ψ_1⟩ dx (Don02, AM19).
    Lemma 3.2; converts total scalar curvature into boundary + Hessian terms.
  • domain assumption Existence of T-invariant cscTW structures (Zhang 2009; Ho 2025 equivariant CR Yamabe).
    Cited in §1.2 to note that existence is already known; the paper’s contribution is the sign analysis and dictionary, not bare existence.
  • domain assumption Transversally Fano toric Sasaki manifolds admit Sasaki–Einstein metrics (Futaki–Ono–Wang 2009).
    Supplies the positive Yamabe side of Corollary 1.6.
  • ad hoc to paper Case-2 vanishing-order hypothesis: ⟨v_T, Hess φ_T v_T⟩ = O(|x−x_T|^α) with α>n−1 along the crease.
    Extra assumption needed for lim Y_CR = −∞ when dim P>1; automatic only for dim N=3.
invented entities (2)
  • Reduced functional I(f) = 2∫_∂P f^{-n} dσ / (∫_P f^{-n-1} dx)^{n/(n+1)} no independent evidence
    purpose: Governs the limiting CR Einstein–Hilbert energy along infinite geodesic ribbons and piecewise-linear test configurations; proposed as the object whose infimum over concave f equals the toric CR Yamabe invariant (Question 5.3).
    Defined in (15) and Prop. 5.1 from the action functional; mathematical construction, not a physical entity.
  • Action functional A^χ_s(t) along toric geodesic ribbons no independent evidence
    purpose: Extends EH to non-smooth piecewise-linear symplectic potentials via ∂_t A, recovering I(f_s).
    Introduced in §5 following LLS23/25; used to prove Prop. 1.7.

pith-pipeline@v1.2.0-daily-grok45 · 42041 in / 3739 out tokens · 71668 ms · 2026-07-31T00:47:35.106253+00:00 · methodology

0 comments
read the original abstract

We study the equivariant CR Yamabe problem for a fixed-point-free torus action on a co-oriented compact contact manifold. Such examples arise naturally in the study of Sasakian manifolds with constant scalar curvature. In the toric case, we show that this problem is equivalent to a kind of boundary value problem for an elliptic PDE on a pair of functions defined on a convex, polyhedral domain. We provide examples of solutions and prove that many contact toric manifolds admit compatible CR structures with distinct signs of CR Yamabe invariants.

Figures

Figures reproduced from arXiv: 2607.27441 by Carlo Scarpa, Christina W. T{\o}nnesen-Friedman, Eveline Legendre.

Figure 1
Figure 1. Figure 1: Paths in the space of symplectic potentials S(P) ⊂ C ∞(P ◦ ) and behaviour of the CR Yamabe function. Theorem 3.7. [[ACMY24, LLS23]] The critical points of EH(φ, f) are characterised by the con￾ditions ( ScalTW(φ, f) = const. Hess(f) = 0. In other words, the critical points are Sasaki structures of constant transversal scalar curvature. 4. The toric CR Yamabe energy In this Section, we prove Theorem 1.4. A… view at source ↗
Figure 2
Figure 2. Figure 2: Graph of EH(cA, cB) when p = 6 and q = 1 [PITH_FULL_IMAGE:figures/full_fig_p022_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Graph of EH(c) when p = 6 and q = 1. • For q > 5p, EHmin = ( 432(p−q) 2 q 2p ) 1/3 . Example 6.2. Suppose p = 6 and q = 1. Then EHmin = (300)1/3 ≈ 6.694. Let A(x) = p 2−x 2 p = 36−x 2 6 and B(y) = q 2−y 2 q = 1−y 2 and consider the corresponding Sasaki contact form η. Evidently, from above we know that η, despite being cscS, is NOT a minimizer of EH over t+, much less over [η] T 2 . As observed in [LLS25],… view at source ↗
Figure 4
Figure 4. Figure 4: Graph of EH(c) when x = 8/10. where g(c) = 24 − 9x − 4x 2 + 12x 3 + 4x 4 − 3x 5 − 8x(5 − 2x 2 + x 4 )c + 2(2 + 3x − 12x 2 − 4x 3 + 2x 4 + x 5 )c 2 + 16x(1 + x 2 )c 3 − (4 − 3x + 4x 2 + 4x 3 − x 5 )c 4 . [As expected EH′ (c) = 0 is equivalent to (31) for s = −3 and a = 3(x 4+7) (1−x2)(3−x2) .] For x = 8/10, we have that EH(c) = 8 5 q 2 15 16437 − 4594c 4 + 16400c 3 − 6533c 2 − 20648c  177(1 − c 2) 4/5 (15 … view at source ↗

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Works this paper leans on

51 extracted references · 1 canonical work pages · 1 internal anchor

  1. [1]

    K\" a hler geometry of toric varieties and extremal metrics

    Miguel Abreu. K\" a hler geometry of toric varieties and extremal metrics. Internat. J. Math. , 9(6):641--651, 1998

    1998

  2. [2]

    K\"ahler- S asaki geometry of toric symplectic cones in action-angle coordinates

    Miguel Abreu. K\"ahler- S asaki geometry of toric symplectic cones in action-angle coordinates. Port. Math. , 67(2):121--153, 2010

    2010

  3. [3]

    Vestislav Apostolov, David M. J. Calderbank, Paul Gauduchon, and Christina W. T nnesen-Friedman. Hamiltonian 2-forms in K \"ahler geometry. II . G lobal classification. J. Differential Geom. , 68(2):277--345, 2004

    2004

  4. [4]

    Vestislav Apostolov, David M. J. Calderbank, Paul Gauduchon, and Christina W. T nnesen-Friedman. Hamiltonian 2-forms in K \"a hler geometry III , extremal metrics and stability. Invent. Math. , 173(3):547--601, 2008

    2008

  5. [5]

    Vestislav Apostolov, David M. J. Calderbank, and Eveline Legendre. Weighted K -stability of polarized varieties and extremality of S asaki manifolds. Adv. Math. , 391:Paper No. 107969, 63, 2021

    2021

  6. [6]

    On the variation of the E instein- H ilbert action in pseudohermitian geometry

    Claudio Afeltra, Jih-Hsin Cheng, Andrea Malchiodi, and Paul Yang. On the variation of the E instein- H ilbert action in pseudohermitian geometry. J. Reine Angew. Math. , 813:81--102, 2024. With an appendix by Xiaodong Wang

    2024

  7. [7]

    Compactness in dimension five and equivariant noncompactness for the CR Y amabe problem

    Claudio Afeltra, Pak Tung Ho, and Andrea Pinamonti. Compactness in dimension five and equivariant noncompactness for the CR Y amabe problem. arXiv:2603.12157 [math.AP] https://arxiv.org/abs/2603.12157v1, 2026

    arXiv 2026

  8. [8]

    Conformally K \"ahler, E instein- M axwell geometry

    Vestislav Apostolov and Gideon Maschler. Conformally K \"ahler, E instein- M axwell geometry. J. Eur. Math. Soc. (JEMS) , 21(5):1319--1360, 2019

    2019

  9. [9]

    The S^1 -equivariant Y amabe invariant of 3-manifolds

    Bernd Ammann, Farid Madani, and Mihaela Pilca. The S^1 -equivariant Y amabe invariant of 3-manifolds. Int. Math. Res. Not. IMRN , (20):6310--6328, 2017

    2017

  10. [10]

    The K \"a hler geometry of toric manifolds

    Vestislav Apostolov. The K \"a hler geometry of toric manifolds. Lecture Notes of CIRM winter school. https://arxiv.org/pdf/2208.12493 arxiv:2208.12493 [math.DG] , 2019

    Pith/arXiv arXiv 2019

  11. [11]

    \'etude d'un certain type d'\'equations diff\'erentielles non lin\'eaires

    Thierry Aubin. \'etude d'un certain type d'\'equations diff\'erentielles non lin\'eaires. C. R. Acad. Sci. Paris S\'er. A-B , 280(7):455--457, 1975

    1975

  12. [12]

    Scalar curvature and isometry group

    Bérard Bergery. Scalar curvature and isometry group. Kaigai Publications , pages 9 -- 28, 1983

    1983

  13. [13]

    Boyer and Krzysztof Galicki

    Charles P. Boyer and Krzysztof Galicki. A note on toric contact geometry. J. Geom. Phys. , 35(4):288--298, 2000

    2000

  14. [14]

    Boyer and Krzysztof Galicki

    Charles P. Boyer and Krzysztof Galicki. Sasakian geometry . Oxford Mathematical Monographs. Oxford University Press, Oxford, 2008

    2008

  15. [15]

    An application of the D uistermaat- H eckman theorem and its extensions in S asaki geometry

    Charles Boyer, Hongnian Huang, and Eveline Legendre. An application of the D uistermaat- H eckman theorem and its extensions in S asaki geometry. Geom. Topol. , 22(7):4205--4234, 2018

    2018

  16. [16]

    Boyer, Hongnian Huang, Eveline Legendre, and Christina W

    Charles P. Boyer, Hongnian Huang, Eveline Legendre, and Christina W. T nnesen Friedman. The E instein- H ilbert functional and the S asaki- F utaki invariant. Int. Math. Res. Not. , (7):1942--1974, 2017

    1942

  17. [17]

    Boyer, Hongnian Huang, Eveline Legendre, and Christina W

    Charles P. Boyer, Hongnian Huang, Eveline Legendre, and Christina W. Tønnesen-Friedman. Some Open Problems in Sasaki Geometry , page 143–168. London Mathematical Society Lecture Note Series. Cambridge University Press, 2020

    2020

  18. [18]

    Boyer, Hongnian Huang, Eveline Legendre, and Christina W

    Charles P. Boyer, Hongnian Huang, Eveline Legendre, and Christina W. T nnesen Friedman. Twins in K \"ahler and S asaki geometry. J. Geom. Phys. , 216:Paper No. 105591, 29, 2025

    2025

  19. [19]

    Banyaga and P

    A. Banyaga and P. Molino. G\'eom\'etrie des formes de contact compl\`etement int\'egrables de type toriques. In S\'eminaire G aston D arboux de G \'eom\'etrie et T opologie D iff\'erentielle, 1991--1992 ( M ontpellier) , pages 1--25. Univ. Montpellier II, Montpellier, 1993

    1991

  20. [20]

    Boyer and Christina W

    Charles P. Boyer and Christina W. T nnesen-Friedman. The S asaki join, H amiltonian 2-forms, and constant scalar curvature. J. Geom. Anal. , 26(2):1023--1060, 2016

    2016

  21. [21]

    Boyer and Christina W

    Charles P. Boyer and Christina W. T nnesen-Friedman. The S^3_w S asaki join construction. J. Math. Soc. Japan , 74(4):1335--1371, 2022

    2022

  22. [22]

    Robustness of CSC Sasaki existence under the join operation

    Charles P. Boyer and Christina W. T nnesen-Friedman. Robustness of CSC S asaki E xistence under the J oin operation. arXiv.2604.09893 [math.DG] https://doi.org/10.48550/arXiv.2604.09893, 2026

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2604.09893 2026

  23. [23]

    Extremal K \"a hler metrics

    Eugenio Calabi. Extremal K \"a hler metrics. In Seminar on Differential Geometry , volume 102 of Annals of Mathematical Studies , pages 259--290. Princeton University Press, 1982

    1982

  24. [24]

    Hamiltoniens p\' e riodiques et images convexes de l'application moment

    Thomas Delzant. Hamiltoniens p\' e riodiques et images convexes de l'application moment. Bull. Soc. Math. France , 116(3):315--339, 1988

    1988

  25. [25]

    Donaldson

    Simon K. Donaldson. Scalar curvature and stability of toric varieties. J. Differential Geom. , 62(2):289--349, 2002

    2002

  26. [26]

    Volume minimization and obstructions to solving some problems in K \"ahler geometry

    Akito Futaki and Hajime Ono. Volume minimization and obstructions to solving some problems in K \"ahler geometry. ICCM Not. , 6(2):51--60, 2018

    2018

  27. [27]

    Transverse K \"ahler geometry of S asaki manifolds and toric S asaki- E instein manifolds

    Akito Futaki, Hajime Ono, and Guofang Wang. Transverse K \"ahler geometry of S asaki manifolds and toric S asaki- E instein manifolds. J. Differential Geom. , 83(3):585--635, 2009

    2009

  28. [28]

    The CR Y amabe conjecture---the case n=1

    Najoua Gamara. The CR Y amabe conjecture---the case n=1 . J. Eur. Math. Soc. (JEMS) , 3(2):105--137, 2001

    2001

  29. [29]

    K \"a hler structures on toric varieties

    Victor Guillemin. K \"a hler structures on toric varieties. J. Differential Geom. , 40(2):285--309, 1994

    1994

  30. [30]

    C R Y amabe conjecture---the conformally flat case

    Najoua Gamara and Ridha Yacoub. C R Y amabe conjecture---the conformally flat case. Pacific J. Math. , 201(1):121--175, 2001

    2001

  31. [31]

    Equivariant CR Y amabe problem

    Pak Tung Ho. Equivariant CR Y amabe problem. Ann. Mat. Pura Appl. (4) , 204(1):289--306, 2025

    2025

  32. [32]

    Hwang and Michael A

    Andrew D. Hwang and Michael A. Singer. A momentum construction for circle-invariant K \"ahler metrics. Trans. Amer. Math. Soc. , 354(6):2285--2325, 2002

    2002

  33. [33]

    Le probl\`eme de Y amabe \'equivariant

    Emmanuel Hebey and Michel Vaugon. Le probl\`eme de Y amabe \'equivariant. Bull. Sci. Math. , 117(2):241--286, 1993

    1993

  34. [34]

    David Jerison and John M. Lee. The Y amabe problem on CR manifolds. J. Differential Geom. , 25(2):167--197, 1987

    1987

  35. [35]

    David Jerison and John M. Lee. Extremals for the S obolev inequality on the H eisenberg group and the CR Y amabe problem. J. Amer. Math. Soc. , 1(1):1--13, 1988

    1988

  36. [36]

    David Jerison and John M. Lee. Intrinsic CR normal coordinates and the CR Y amabe problem. J. Differential Geom. , 29(2):303--343, 1989

    1989

  37. [37]

    K\"ahler metrics with constant weighted scalar curvature and weighted K -stability

    Abdellah Lahdili. K\"ahler metrics with constant weighted scalar curvature and weighted K -stability. Proc. Lond. Math. Soc. (3) , 119(4):1065--1114, 2019

    2019

  38. [38]

    The E instein- M axwell equations, K \"ahler metrics, and H ermitian geometry

    Claude LeBrun. The E instein- M axwell equations, K \"ahler metrics, and H ermitian geometry. J. Geom. Phys. , 91:163--171, 2015

    2015

  39. [39]

    Yamabe invariants, homogeneous spaces, and rational complex surfaces

    Claude LeBrun. Yamabe invariants, homogeneous spaces, and rational complex surfaces. SIGMA Symmetry Integrability Geom. Methods Appl. , 19:Paper No. 027, 11, 2023

    2023

  40. [40]

    Existence and non-uniqueness of constant scalar curvature toric S asaki metrics

    Eveline Legendre. Existence and non-uniqueness of constant scalar curvature toric S asaki metrics. Compos. Math. , 147(5):1613--1634, 2011

    2011

  41. [41]

    Contact toric manifolds

    Eugene Lerman. Contact toric manifolds. J. Symplectic Geom. , 1(4):785--828, 2003

    2003

  42. [42]

    The E instein- H ilbert functional and the D onaldson- F utaki I nvariant

    Abdellah Lahdili, Eveline Legendre, and Carlo Scarpa. The E instein- H ilbert functional and the D onaldson- F utaki I nvariant. arXiv:2310.11625 [math.DG] https://arxiv.org/abs/2310.11625, 2023

    Pith/arXiv arXiv 2023

  43. [43]

    The CR Y amabe invariant and constant scalar curvature S asaki metrics

    Abdellah Lahdili, Eveline Legendre, and Carlo Scarpa. The CR Y amabe invariant and constant scalar curvature S asaki metrics. arXiv:2509.00743 [math.DG] https://arxiv.org/abs/2509.00743, 2025

    Pith/arXiv arXiv 2025

  44. [44]

    Hamiltonian torus actions on symplectic orbifolds and toric varieties

    Eugene Lerman and Susan Tolman. Hamiltonian torus actions on symplectic orbifolds and toric varieties. Trans. Amer. Math. Soc. , 349(10):4201--4230, 1997

    1997

  45. [45]

    An O bata-type theorem in CR geometry

    Song-Ying Li and Xiaodong Wang. An O bata-type theorem in CR geometry. J. Differential Geom. , 95(3):483--502, 2013

    2013

  46. [46]

    Equivariant Y amabe problem and H ebey- V augon conjecture

    Farid Madani. Equivariant Y amabe problem and H ebey- V augon conjecture. J. Funct. Anal. , 258(1):241--254, 2010

    2010

  47. [47]

    Hebey- V augon conjecture II

    Farid Madani. Hebey- V augon conjecture II . C. R. Math. Acad. Sci. Paris , 350(17-18):849--852, 2012

    2012

  48. [48]

    The geometric dual of a -maximisation for toric S asaki- E instein manifolds

    Dario Martelli, James Sparks, and Shing-Tung Yau. The geometric dual of a -maximisation for toric S asaki- E instein manifolds. Comm. Math. Phys. , 268(1):39--65, 2006

    2006

  49. [49]

    Real and complex analysis

    Walter Rudin. Real and complex analysis . McGraw-Hill Book Co., New York-Toronto-London, 1966

    1966

  50. [50]

    The CR Y amabe constant and inequivalent CR structures

    Chanyoung Sung and Yuya Takeuchi. The CR Y amabe constant and inequivalent CR structures. Pacific J. Math. , 335(1):163--181, 2025

    2025

  51. [51]

    The contact Y amabe flow on K -contact manifolds

    YongBing Zhang. The contact Y amabe flow on K -contact manifolds. Sci. China Ser. A , 52(8):1723--1732, 2009

    2009