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The sharp $\sigma_2$-curvature inequality on the sphere in quantitative form

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Pith's one-line read Near-minimizers of the total σ2-curvature on the sphere are quantitatively close to Möbius images of the round metric.

desk verdict First quantitative stability theorem for a fully nonlinear conformal curvature functional, with optimal two-norm behavior; proof is solid, but referee should verify the external Ge-Wang inequality that carries the global-to-local step. read the letter →

arxiv 2412.12819 v1 pith:VZYPSDGI submitted 2024-12-17 math.AP math.DGmath.FA

Rupert L. Frank , Jonas W. Peteranderl This is my paper · ORCID
classification math.APmath.DGmath.FA MSC 35A2335J6053C21
keywords σ2-curvatureconformalgeometrysharpinequalitystabilitySobolevMöbiustransformationsfullynonlinearsphere
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 a quantitative stability version of the sharp σ2-curvature inequality on the sphere S^d for d>4. It shows that any conformal metric with positive scalar curvature whose normalized total σ2-curvature is close to the minimum must be close, in explicit Sobolev norms, to a Möbius image of the standard metric. The two exponents, 2 and 4, are shown to be optimal. This is the first stability result of this kind for a variational problem whose Euler–Lagrange equation is fully nonlinear.

What carries the argument

The proof is built on three pieces. The first is the quotient inequality $(F_2/S_d^{(2)})^{1/(d-4)} \ge (F_1/S_d^{(1)})^{1/(d-2)}$, imported from the literature, which transfers near-optimality of F_2 to near-optimality of the Yamabe functional F_1; compactness for F_1-optimizing sequences then comes from concentration compactness. The second is the nonnegative energy density e_2(u) (defined in terms of σ1(u), |∇u|^2, and u), which turns the total σ2-curvature into an integral of nonnegative terms and permits a Taylor expansion around the optimizer. The third is a decomposition of the remainder r=u-1 into spherical harmonics of low, medium, and high degree; the medium frequencies are finite-dimensional and can be pushed into higher-order errors, while the high frequencies are where the quartic $W^{{1,4}}$ terms contribute at the sharp order, giving the exponent 4.

What would settle it

Two concrete tests: (i) compute the quotient ratio in (2.1) for metrics with σ1>0 but σ2≤0 to see if it fails; (ii) try to build an optimizing sequence with F_2[u]→$S_d^{{(2)}}$ whose $W^{{1,4}}$-distance to the Möbius orbit stays bounded away from zero. Either violation would falsify the main stability claim.

Watch

Extended reading notes

Core claim

The central assertion is Theorem 1: for every d>4 there is a constant c_d>0 such that every positive smooth function u on S^d with σ1(u)>0 satisfies F_2[u]-$S_d^{{(2)}}$ ≥ c_d inf_{λ,Ψ} ( ‖λ(u)_Ψ-1‖^2_{$W^{{1,2}}$} + ‖λ(u)_Ψ-1‖^4_{$W^{{1,4}}$} ), where F_2 is the normalized total σ2-curvature functional, $S_d^{{(2)}}$ its sharp minimum, and (u)_Ψ is u transformed by the Möbius map Ψ with the appropriate Jacobian weight. In geometric terms, almost minimizers of the total σ2-curvature among unit-volume conformal metrics with positive scalar curvature are quantitatively close to the standard metric and its Möbius images. The closeness is strong enough to control Sobolev distances by the energy deficit, and the exponents 2 and 4 cannot be improved, as shown by explicit families of perturbations.

Load-bearing premise

The whole compactness reduction rests on a quoted inequality comparing the σ2-functional with the scalar-curvature functional for all metrics of positive scalar curvature; if that inequality were not true for all such metrics, the proof's first step would fail.

Editorial extensions

If this is right

  • The deficit F_2[u]-S_d^{(2)} controls both the quadratic W^{1,2} distance and the quartic W^{1,4} distance to the Möbius orbit, so any convergence of the functional implies convergence of the metric in those norms.
  • The optimality results mean that no stability inequality of the same form can hold with a smaller power than 2 for the W^{1,2} distance or smaller than 4 for the W^{1,4} distance, settling the sharp order of the remainder.
  • Optimizing sequences for F_2 are relatively compact in W^{1,4} modulo Möbius transformations, giving a compactness theorem for the fully nonlinear problem that does not require a direct concentration-compactness argument for F_2 itself.
  • The local analysis around minimizers yields a two-term stability inequality for the classical Sobolev inequality in \dot W^{1,p}(R^d) with 2<p<d, strengthening the existing gradient stability result in that setting.

Reading between the lines

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

  • This suggests that the quantitative distance bound could serve as the key input for controlling convergence rates of gradient flows for the fully nonlinear σ2-curvature functional, even though no such flow result is proved here.
  • The method of transferring compactness via the quotient inequality might adapt to other σ_k-curvature functionals on the sphere, provided an analogue of the quotient inequality and a nonnegative energy density exist, potentially yielding stability for a whole family of fully nonlinear variational problems.
  • The explicit spectral decomposition raises the possibility of computing explicit stability constants in Theorem 1 by tracking the spectral gap and the constants in the quotient inequality; the paper notes this is conceivable but does not carry it out.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 6 minor

Summary. The paper establishes a quantitative (Bianchi–Egnell type) stability theorem for the sharp σ2-curvature inequality on the sphere. For d>4, any positive smooth conformal factor u with σ1(u)>0 satisfies a deficit estimate in terms of the best W^{1,2} and W^{1,4} distance to the Möbius orbit of the constant function, with the optimal powers 2 and 4 respectively. The proof consists of a global-to-local reduction (Proposition 3) based on the Ge–Wang quotient monotonicity inequality (2.1) and Lions concentration compactness, followed by a local analysis (Proposition 4) using a spherical harmonic decomposition and a careful Taylor expansion. Sharpness of the two exponents is demonstrated by two distinct families of examples in Section 5. An appendix gives a strengthening of the Figalli–Zhang stability inequality for the p-Sobolev inequality.

Significance. This is the first quantitative stability result for a conformal variational problem whose Euler–Lagrange equation is fully nonlinear, which is a substantive advance over the existing stability theory for the Yamabe/Sobolev inequalities. The main theorem is sharp in the sense that the exponents 2 and 4 cannot be improved, and the sharpness constructions are explicit. The proof is technically original: the use of the Ge–Wang monotonicity formula to transfer compactness from F1 to F2, the two-norm local analysis via spherical harmonics, and the careful handling of the pointwise constraint σ1>0 are all strong points. The paper is largely self-contained except for the clearly identified external inputs (2.1) from [GW04] and [GW13], and the auxiliary result in the appendix is a nice complement. The central argument appears sound; the issues I found are local and repairable.

major comments (2)
  1. [§2, proof of Proposition 3] Just after defining sj and rj, the paper states: 'By the Sobolev inequality, we have sj → 0 in L^{2d/(d−2)} and therefore also rj → 0 in L^{4d/(d−4)} (Sd).' This inference is not valid as written: for d>4, L^{4d/(d−4)} is a stronger space than L^{2d/(d−2)} on a finite measure space, and convergence in W^{1,2} only gives convergence in L^{2d/(d−2)}. Nevertheless, the needed conclusion (1+rj)^4 → 1 in L^1 follows by Vitali's theorem, since rj → 0 in measure (because sj → 0 in W^{1,2}) and ||1+rj||_{L^{4d/(d−4)}} is bounded by the normalization. I recommend replacing the incorrect Sobolev-embedding statement with this standard uniform-integrability argument.
  2. [Appendix, proof of Theorem 14] In the case δ(u) > δ0 ||∇u||^p, the proof claims the bound inf_{Q∈M} ( ||∇u−∇Q||^p + ∫|∇Q|^{p−2}|∇u−∇Q|^2 dx ) ≤ ||∇u||^p 'by taking Q=0'. But Q=0 is not an optimizer, so it is not admissible in the infimum over M. This step needs a correct competitor (for example, a sequence of optimizers concentrating at infinity, which would give a bound by a constant times ||∇u||^p), or a different argument. Since this is the only proof of the non-degenerate regime, the appendix is incomplete as written.
minor comments (6)
  1. [§4.1, definition of frequency decomposition] In the decomposition after Lemma 12, the third line reads 'rmed_j := ...', which is a typo; it should define rhi_j.
  2. [§4.1, inequalities (4.3) and (4.4)] The coefficient in the quadratic term is printed as 2(3d+4)/(d−4) in (4.3) but as 2(3d−4)/(d−4) in (4.4); one of these is a typo.
  3. [§5.2, equations (5.6) and (5.7)] The notation 'o_{|ξ|→∞}(1)' appears where the limit should be as |ξ|→1; please correct to 'o_{|ξ|→1}(1)'.
  4. [§2, inequality (2.1)] Since inequality (2.1) is the pivotal external input for Proposition 3, the authors should state explicitly which theorem of [GW04] and which theorem of [GW13] are used and confirm that the hypotheses of those results match the present setting (in particular, that the removal of the σ2>0 condition is exactly as in [GW13]). This is a clarity request, not a doubt about correctness.
  5. [§1.1, conformal parametrization] The sentence 'the functional F1 attains its minimum precisely at those metrics that are obtained from g∗ by a Möbius transformation' would be more precise as '...at the metrics Ψ*g∗ for Möbius transformations Ψ', since the set of minimizers is the orbit, not a single metric.
  6. [§4.2, proof of Lemma 13, Part 1] In the step estimating Π1rj, the factor (d+1)/|Sd| appears in the projection formula; the subsequent bound is correct, but it would be helpful to state that this constant depends only on d.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the quantitative stability theorem is proved from an external sharp inequality input, with no fitted parameters or self-citation chain forcing the result.

full rationale

The paper's target result, Theorem 1, is a quantitative stability statement for the sharp σ2-curvature inequality (1.6). The proof does not assume the theorem. The central global-to-local reduction (Proposition 3) relies on the quotient inequality (2.1), which is explicitly cited as an external result from Guan–Wang [GW04, Theorem 1] and Ge–Wang [GW13, Theorem 1], and the paper explains the two-step deduction. This is a normal use of prior mathematical results, not a circular reduction: inequality (2.1) is not equivalent to the stability conclusion and does not contain the W^{1,2}/W^{1,4} distance terms being proved. The local analysis (Proposition 4) is carried out from the explicit energy density (1.5), expansions, spherical harmonic decompositions, and elementary estimates; no parameter is fitted to the data being predicted. Self-citations ([Fra22], [Fra24], [FP24]) appear only for Möbius parametrizations, Hardy-type bounds, and related stability models, and none of these carries the argument by itself; the key compactness comes from Lions's theorem and the cited quotient inequality. The appendix extends a result of Figalli–Zhang [FZ22] using existing bounds from [FN19], again as external input. There is no step where the theorem's conclusion is used as an assumption, no fitted quantity is renamed as a prediction, and no uniqueness claim is imported solely from the authors' own prior work. The only potential concern is whether the cited Ge–Wang quotient inequality holds under exactly the stated hypotheses, which is a correctness or external-support question, not a circularity one.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central claim rests on standard external theorems (Ge-Wang quotient inequality, Lions concentration compactness, Case's energy identity, Hardy inequality) and on the domain assumption σ1(u)>0. There are no free parameters fitted to data and no new postulated entities. The proof derives the quantitative inequality rather than assuming it.

assumptions (7)
  • standard math Sharp quotient inequality (2.1): (F2/S2)^(1/(d-4)) ≥ (F1/S1)^(1/(d-2)) for metrics with σ1>0.
    Invoked in Proposition 3 to turn F2 near-minimality into F1 near-minimality; cited to Guan-Wang [GW04] and Ge-Wang [GW13]. Not proved in this paper.
  • standard math Lions concentration-compactness principle for optimizing sequences of the sharp Sobolev inequality on S^d.
    Used in Proposition 3 to get convergence of an F1-optimizing sequence after Möbius transformations; cited [Lio85a, Lio85b].
  • standard math Case's energy-density identity ∫ σ2 = ∫ e2(u), with e2(u) as in (1.5).
    External identity from [Cas20]; converts the geometric functional into an analytic functional with nonnegative density.
  • standard math Hardy inequality ∫ |v|^4/|x|^4 dx ≲ ∫ |∇v|^4 dx on R^d, d>4.
    Used in the proof of Lemma 6 to control the singularity of conformal images in W^1,4.
  • standard math Norm equivalence of W^1,2 and W^1,4 on finite-dimensional spherical harmonic spaces of bounded degree.
    Used throughout Section 4.3 to discard low- and medium-frequency remainders.
  • standard math Conformal invariance of F2 under Möbius transformations and invariance of the sharp inequality (1.6).
    Used throughout; follows from the conformal transformation formulas for σ2-curvature.
  • domain assumption The theorem restricts to u with σ1(u)>0.
    The positivity of σ1(u) is part of the theorem hypothesis and is used to discard nonnegative terms, e.g., after (4.11), where σ1(1+r_j) > 0 is applied.

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Pith. "Pith review of The sharp $\sigma_2$-curvature inequality on the sphere in quantitative form." pith.science (2026). https://pith.science/paper/VZYPSDGI

@misc{pith2026241212819,
  author       = {Pith},
  title        = {Pith review of: The sharp $\sigma_2$-curvature inequality on the sphere in quantitative form},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VZYPSDGI}},
  note         = {Machine review of arXiv:2412.12819}
}
abstract

Among all metrics on $\mathbb S^d$ with $d>4$ that are conformal to the standard metric and have positive scalar curvature, the total $\sigma_2$-curvature, normalized by the volume, is uniquely (up to M\"obius transformations) minimized by the standard metric. We show that if a metric almost minimizes, then it is almost the standard metric (up to M\"obius transformations). This closeness is measured in terms of Sobolev norms of the conformal factor, and we obtain the optimal stability exponents for two different notions of closeness. This is a stability result for an optimization problem whose Euler-Lagrange equation is fully nonlinear.

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Reference graph

Works this paper leans on

46 extracted references · 45 canonical work pages · cited by 2 Pith papers

  1. [1]

    Aubin, Equations diff\'erentielles non lin\'eaires et probl\`eme de Y amabe concernant la courbure scalaire , J

    T. Aubin, Equations diff\'erentielles non lin\'eaires et probl\`eme de Y amabe concernant la courbure scalaire , J. Math. Pures Appl. 55 (1976), 269--296

    work page 1976

  2. [2]

    Bianchi and H

    G. Bianchi and H. Egnell, A note on the sobolev inequality, J. Funct. Anal. 100 (1991), no. 1, 18--24

    work page 1991

  3. [3]

    Bonforte, J

    M. Bonforte, J. Dolbeault, B. Nazaret, and N. Simonov, Constructive stability results in interpolation inequalities and explicit improvements of decay rates of fast diffusion equations, Discrete Contin. Dyn. Syst. 43 (2023), no. 3 & 4, 1070--1089

    work page 2023

  4. [4]

    Bonforte, J

    M. Bonforte, J. Dolbeault, B. Nazaret, and N. Simonov, Stability in gagliardo-nirenberg-sobolev inequalities: Flows, regularity and the entropy method, Memoirs of the AMS, to appear (2025), arXiv:2007.03674

    arXiv 2025

  5. [5]

    Brendle, Convergence of the Y amabe flow for arbitrary initial energy , J

    S. Brendle, Convergence of the Y amabe flow for arbitrary initial energy , J. Differential Geom. 69 (2005), no. 2, 217--278

    work page 2005

  6. [6]

    Brendle, Convergence of the Y amabe flow in dimension 6 and higher , Invent

    S. Brendle, Convergence of the Y amabe flow in dimension 6 and higher , Invent. Math. 170 (2007), no. 3, 541--576

    work page 2007

  7. [7]

    Brezis and E

    H. Brezis and E. H. Lieb, Sobolev inequalities with remainder terms, J. Funct. Anal. 62 (1985), no. 1, 73--86

    work page 1985

  8. [8]

    Brigati, J

    G. Brigati, J. Dolbeault, and N. Simonov, Logarithmic S obolev and interpolation inequalities on the sphere: constructive stability results , Ann. Inst. H. Poincar\'e C Anal. Non Lin\'eaire 41 (2024), no. 5, 1289--1321

    work page 2024

Show all 46 references
  1. [9]

    Carlotto, O

    A. Carlotto, O. Chodosh, and Y. A. Rubinstein, Slowly converging yamabe flows, Geom. Topol. 19 (2015), no. 3, 1523--1568

    2015

  2. [10]

    J. S. Case, The Frank-Lieb approach to sharp S obolev inequalities , Commun. Contemp. Math. 23 (2020), no. 03, 2050015

    2020

  3. [11]

    J. S. Case and Y. Wang, Towards a fully nonlinear sharp sobolev trace inequality, J. Math. Study 53 (2020), no. 4, 402--435

    2020

  4. [12]

    Cianchi, N

    A. Cianchi, N. Fusco, F. Maggi, and A. Pratelli, The sharp S obolev inequality in quantitative form , J. Eur. Math. Soc. 11 (2009), no. 5, 1105--1139

    2009

  5. [13]

    S.-Y. A. Chang, M. J. Gursky, and P. C. Yang, An a priori estimate for a fully nonlinear equation on four-manifolds, J. Anal. Math. 87 (2002), no. 1, 151--186

    2002

  6. [14]

    S.-Y. A. Chang, M. J. Gursky, and P. C. Yang, An Equation of Monge-Ampere Type in Conformal Geometry, and Four-Manifolds of Positive Ricci Curvature , Ann. Math. 155 (2002), no. 3, 709--787

    2002

  7. [15]

    S. Chen, R. L. Frank, and T. Weth, Remainder T erms in the F ractional S obolev I nequality , Indiana Univ. Math. J. 62 (2013), no. 4, 1381--1397

    2013

  8. [16]

    Dolbeault, M

    J. Dolbeault, M. J. Esteban, A. Figalli, R. L. Frank, and M. Loss, Sharp stability for S obolev and log- S obolev inequalities, with optimal dimensional dependence , Preprint (2023), arXiv:2209.08651

    2023 arXiv

  9. [17]

    Engelstein, R

    M. Engelstein, R. Neumayer, and L. Spolaor, Quantitative stability for minimizing Y amabe metrics , Trans. Amer. Math. Soc. Ser. B 9 (2022), 395--414

    2022

  10. [18]

    Figalli and R

    A. Figalli and R. Neumayer, Gradient stability for the S obolev inequality: the case p 2 , J. Eur. Math. Soc. 29 (2019), no. 2, 319--354

    2019

  11. [19]

    Figalli and Y

    A. Figalli and Y. R.-Y. Zhang, Sharp gradient stability for the S obolev inequality , Duke Math. J. 171 (2022), no. 12, 2407--2459

    2022

  12. [20]

    R. L. Frank, Degenerate stability of some S obolev inequalities , Ann. Inst. H. Poincaré Anal. Non Linéaire 39 (2022), no. 6, 1459--1484

    2022

  13. [21]

    R. L. Frank, The sharp S obolev inequality and its stability: An introduction , Geometric and Analytic Aspects of Functional Variational Principles. Cetraro, Italy 2022 (A. Cianchi, V. Maz'ya, and T. Weth, eds.), Springer Cham, 2024, 1--64

    2022

  14. [22]

    R. L. Frank and E. H. Lieb, A new, rearrangement-free proof of the sharp hardy-littlewood-sobolev inequality, Spectral Theory, Function Spaces and Inequalities (B. Malcolm Brown, Jan Lang, and Ian G. Wood, eds.), Springer Basel, 2012, 55--67

    2012

  15. [23]

    R. L. Frank and E. H. Lieb, Sharp constants in several inequalities on the H eisenberg group , Ann. of Math. 176 (2012), no. 1, 349--381

    2012

  16. [24]

    R. L. Frank and J. W. Peteranderl, Degenerate stability of the C affarelli- K ohn- N irenberg inequality along the F elli- S chneider curve , Calc. Var. Partial Differential Equations 63 (2024), no. 44

    2024

  17. [25]

    Ge and G

    Y. Ge and G. Wang, On a fully nonlinear Y amabe problem , Ann. Sci. \'Ec. Norm. Sup\'er. 39 (2005), 569--598

    2005

  18. [26]

    Ge and G

    Y. Ge and G. Wang, On a conformal quotient equation. II , Commun. Anal. Geom. 21 (2013), no. 1, 1--38

    2013

  19. [27]

    P. Guan, J. Viaclovsky, and G. Wang, Some Properties of the Schouten Tensor and Applications to Conformal Geometry , Trans. Amer. Math. Soc. 355 (2003), no. 3, 925--933

    2003

  20. [28]

    Guan and G

    P. Guan and G. Wang, Geometric inequalities on locally conformally flat manifolds, Duke Math. J. 124 (2004), no. 1, 177--212

    2004

  21. [29]

    Guerra, X

    A. Guerra, X. Lamy, and K. Zemas, Sharp quantitative stability of the M \"obius group among sphere-valued maps in arbitrary dimension , Trans. Amer. Math. Soc., to appear (2023), arXiv:2305.19886

    2023 arXiv

  22. [30]

    M. J. Gursky and J. A. Viaclovsky, Prescribing Symmetric Functions of the Eigenvalues of the Ricci Tensor , Ann. Math. 166 (2007), no. 2, 475--531

    2007

  23. [31]

    K \"o nig, An exceptional property of the one-dimensional bianchi--egnell inequality, Calc

    T. K \"o nig, An exceptional property of the one-dimensional bianchi--egnell inequality, Calc. Var. Partial Differential Equations 63 (2024), no. 123

    2024

  24. [32]

    J. M. Lee and T. H. Parker, The Y amabe problem , Bull. Amer. Math. Soc. (N.S.) 17 (1987), no. 1, 37--91

    1987

  25. [33]

    Li and Y

    A. Li and Y. Y. Li, On some conformally invariant fully nonlinear equations, Commun. Pure Appl. Math. 56 (2003), no. 10, 1416--1464

    2003

  26. [34]

    Lions, The C oncentration- C ompactness P rinciple in the C alculus of V ariations

    P.-L. Lions, The C oncentration- C ompactness P rinciple in the C alculus of V ariations. T he limit case, P art 1. , Rev. Mat. Iberoam. 1 (1985), no. 1, 145--201

    1985

  27. [35]

    Lions, The C oncentration- C ompactness P rinciple in the C alculus of V ariations

    P.-L. Lions, The C oncentration- C ompactness P rinciple in the C alculus of V ariations. T he limit case, P art 2. , Rev. Mat. Iberoam. 1 (1985), no. 2, 45--121

    1985

  28. [36]

    Neumayer, A note on strong-form stability for the S obolev inequality , Calc

    R. Neumayer, A note on strong-form stability for the S obolev inequality , Calc. Var. Partial Differential Equations 59 (2020), no. 25

    2020

  29. [37]

    Obata, The conjectures on conformal transformations of R iemannian manifolds , J

    M. Obata, The conjectures on conformal transformations of R iemannian manifolds , J. Differential Geom. 6 (1971), no. 2, 247--258

    1971

  30. [38]

    Rodemich, The S obolev inequality with best possible constant , Analysis Seminar Caltech (1966)

    E. Rodemich, The S obolev inequality with best possible constant , Analysis Seminar Caltech (1966)

    1966

  31. [39]

    R. M. Schoen, Conformal deformation of a R iemannian metric to constant scalar curvature , J. Differential Geom. 20 (1984), 479--495

    1984

  32. [40]

    Sheng, N

    W.-M. Sheng, N. S. Trudinger, and X.-J. Wang, The Y amabe problem for higher order curvatures , J. Differential Geom. 77 (2007), no. 3, 515--553

    2007

  33. [41]

    Sheng, N

    W.-M. Sheng, N. S. Trudinger, and X.-J. Wang, The k- Y amabe problem , International Press (2012), 427--457

    2012

  34. [42]

    E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces. Sixth Printing , Princeton University Press, 1990

    1990

  35. [43]

    Talenti, B est constant in S obolev inequality , Ann

    G. Talenti, B est constant in S obolev inequality , Ann. Mat. Pura Appl. 110 (1976), no. 1, 353--372

    1976

  36. [44]

    N. S. Trudinger, Remarks concerning the conformal deformation of riemannian structures on compact manifolds, Ann. Sc. Norm. Super. Pisa - Scienze Fisiche e Mat. 22 (1968), no. 2, 265--274

    1968

  37. [45]

    J. A. Viaclovsky, Conformal geometry, contact geometry, and the calculus of variations, Duke Math. J. 101 (2000), no. 2, 283--316

    2000

  38. [46]

    Yamabe, On a deformation of R iemannian structures on compact manifolds , Osaka Math

    H. Yamabe, On a deformation of R iemannian structures on compact manifolds , Osaka Math. J. 12 (1960), no. 1, 21--37

    1960

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