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arxiv: 2605.05414 · v2 · pith:RWTQNISAnew · submitted 2026-05-06 · 🧮 math.DG · math.AP

The σ_k-Yamabe problem revisited

Yuxin Ge , Guofang Wang , Wei Wei This is my paper

Pith reviewed 2026-05-20 23:03 UTC · model grok-4.3

classification 🧮 math.DG math.AP
keywords σk-Yamabe problemconformal metricsYamabe constantscalar curvatureσ2 curvatureclosed manifoldspositive curvature
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The pith

On manifolds with positive Yamabe constant, a positive σ2-Yamabe constant is achieved by a conformal metric with positive scalar curvature.

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

The paper proves that on a closed manifold with positive first Yamabe constant, if the σ2-Yamabe constant is also positive, then this constant is attained by a metric in the given conformal class that has positive scalar curvature. Attaining the constant produces a metric of constant σ2-curvature and therefore solves the σ2-Yamabe problem. The same positivity assumptions imply that the infimum of the normalized σ2 integral over positive-scalar-curvature metrics equals the infimum taken over the further restricted set where σ2 is positive as well. The authors show that both the achievement and the equality of infima cease to hold once the positive scalar curvature restriction is dropped.

Core claim

On a closed manifold (M, [g0]) with Y1(M, [g0]) > 0, if Y2(M, [g0]) > 0, then the σ2-Yamabe constant Y2 is achieved by some conformal metric g in [g0] satisfying R_g > 0. This metric has constant σ2-curvature and therefore solves the σ2-Yamabe problem. As a consequence, the infimum of the functional over all metrics with R_g > 0 equals the infimum over the subset of those metrics that additionally satisfy σ2(g) > 0.

What carries the argument

The σ2-Yamabe constant Y2, defined as the infimum of the integral of σ2(g) dvol(g) divided by vol(g) raised to the power (n-4)/n, taken over all conformal metrics g with positive scalar curvature R_g > 0.

If this is right

  • The σ2-Yamabe problem admits a solution whenever both Yamabe constants are positive.
  • The infimum of the functional over positive-scalar-curvature metrics coincides with the infimum over the subset where σ2 is also positive.
  • Both the existence and the equality of infima fail in general if the positive scalar curvature condition is omitted.

Where Pith is reading between the lines

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

  • The emphasis on positivity of scalar curvature may indicate that similar restrictions could be useful when studying higher-order σk problems.
  • The result raises the possibility that the same variational approach could be tested on manifolds where only partial positivity conditions hold.
  • One could look for explicit examples of manifolds where Y2 is positive yet the minimizing metric has constant σ2 equal to a specific value determined by topology.

Load-bearing premise

The requirement that scalar curvature stay positive for the metrics considered is essential for both the achievement of the infimum and the equality of the two infima.

What would settle it

A closed manifold with Y1 > 0 and Y2 > 0 on which no conformal metric with positive scalar curvature attains the value of Y2 would disprove the main existence result.

read the original abstract

In this paper we revisit the $\sigma_k$-Yamabe problem on $M^n$, namely, finding a conformal metric with constant $\sigma_k$-scalar curvature. We prove that on a closed manifold $\left(M,\left[g_0\right]\right)$ with positive Yamabe constant $Y_1\left(M,\left[g_0\right]\right)>0$, the $\sigma_2$-Yamabe constant $$ Y_2\left(M,\left[g_0\right]\right):=\inf _{g \in\left[g_0\right], R_g>0} \frac{\int_M \sigma_2(g) d \operatorname{vol}(g)}{\operatorname{vol}(g)^{\frac{n-4}{n}}} $$ is achieved by a conformal metric $g \in\left[g_0\right]$, which in particular solves the $\sigma_2$-Yamabe problem, assuming $Y_2\left(M,\left[g_0\right]\right)>0$. As a consequence, for any $\left(M, g_0\right)$ with $Y_1\left(M,\left[g_0\right]\right)>$ 0 and $Y_2\left(M,\left[g_0\right]\right)>0$ one has $$ \inf _{g \in\left[g_0\right], R_g>0} \frac{\int_M \sigma_2(g) d \operatorname{vol}(g)}{\operatorname{vol}(g)^{\frac{n-4}{n}}}=\inf _{g \in\left[g_0\right], R_g>0, \sigma_2(g)>0} \frac{\int_M \sigma_2(g) d \operatorname{vol}(g)}{\operatorname{vol}(g)^{\frac{n-4}{n}}} . $$ We also show that these conclusions can fail if the condition $R_g>0$ is removed.

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 / 3 minor

Summary. The paper revisits the σ₂-Yamabe problem on closed manifolds. It proves that if Y₁(M,[g₀])>0 and Y₂(M,[g₀])>0, then the σ₂-Yamabe constant Y₂(M,[g₀]), defined as the infimum of ∫_M σ₂(g) dvol(g) / vol(g)^{(n-4)/n} over conformal metrics g∈[g₀] with R_g>0, is achieved by some g∈[g₀]. This yields a solution to the σ₂-Yamabe equation with constant positive σ₂. As a corollary the two infima (over R_g>0 and over R_g>0 with σ₂(g)>0) coincide. Counterexamples are given showing both conclusions fail without the R_g>0 restriction.

Significance. The result supplies a clean existence theorem for the σ₂-Yamabe problem under the natural positivity hypotheses Y₁>0 and Y₂>0, together with an explicit demonstration that the scalar-curvature positivity constraint is necessary. The direct-method argument and the accompanying counterexamples constitute a useful contribution to the literature on fully nonlinear Yamabe-type problems in conformal geometry.

minor comments (3)
  1. The introduction should contain a short paragraph summarizing the structure of the direct-method argument (minimizing sequence, compactness, positivity preservation) so that the reader can follow the logic before the technical sections.
  2. In the statement of the main theorem, explicitly record that the limiting metric satisfies σ₂(g)>0 (which follows from the Euler-Lagrange equation once the infimum is achieved).
  3. A brief comparison with the known k=1 (Yamabe) case and with existing partial results for k=2 would help situate the new contribution.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive report and the recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The central result defines Y₂(M,[g₀]) explicitly as the infimum of the normalized ∫σ₂(g) dvol(g) over the subset of conformal metrics with R_g > 0, then applies a direct-method variational argument to produce a minimizer that achieves this infimum and satisfies the Euler-Lagrange equation with constant positive σ₂. The equality between the two infima is an immediate corollary of this achievement. The paper separately exhibits counter-examples showing both conclusions fail without the R_g > 0 restriction, confirming the hypothesis is substantive. No step reduces by construction to a fitted parameter, self-referential definition, or load-bearing self-citation; the argument is self-contained within standard variational compactness and elliptic regularity techniques.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard background from differential geometry and elliptic PDE theory; no free parameters or invented entities are introduced in the abstract statement.

axioms (2)
  • standard math M is a closed smooth manifold of dimension n
    Standard setting for the Yamabe-type problems on compact manifolds without boundary.
  • domain assumption The conformal class [g0] admits metrics with positive scalar curvature when Y1>0
    The positive Yamabe constant assumption is used to restrict to the positive scalar curvature cone.

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Works this paper leans on

34 extracted references · 34 canonical work pages

  1. [1]

    Thierry Aubin,équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire, J. Math. Pures Appl. (9)55(1976), no. 3, 269–296. MR 431287

    work page 1976

  2. [2]

    Math.170(2007), no

    Simon Brendle,Convergence of the Yamabe flow in dimension 6 and higher, Invent. Math.170(2007), no. 3, 541–576. MR 2357502

    work page 2007

  3. [3]

    Math.223(2010), no

    Giovanni Catino and Zindine Djadli,Conformal deformations of integral pinched 3-manifolds, Adv. Math.223(2010), no. 2, 393–404. MR 2565533

    work page 2010

  4. [4]

    Chang, Matthew Gursky, and Siyi Zhang,A conformally invariant gap theorem charac- terizingCP 2 via the Ricci flow, Math

    Sun-Yung A. Chang, Matthew Gursky, and Siyi Zhang,A conformally invariant gap theorem charac- terizingCP 2 via the Ricci flow, Math. Z.294(2020), no. 1-2, 721–746. MR 4050082

    work page 2020

  5. [5]

    Chang, Matthew J

    Sun-Yung A. Chang, Matthew J. Gursky, and Paul C. Yang,An equation of Monge-Ampère type in conformal geometry, and four-manifolds of positive Ricci curvature, Ann. of Math. (2)155(2002), no. 3, 709–787. MR 1923964

    work page 2002

  6. [6]

    ,A conformally invariant sphere theorem in four dimensions, Publ. Math. Inst. Hautes Études Sci. (2003), no. 98, 105–143. MR 2031200

    work page 2003

  7. [7]

    Yang,The inequality of Moser and Trudinger and applications to conformal geometry, Comm

    Sun-Yung Alice Chang and Paul C. Yang,The inequality of Moser and Trudinger and applications to conformal geometry, Comm. Pure Appl. Math.56(2003), no. 8, 1135–1150, Dedicated to the memory of Jürgen K. Moser. MR 1989228

    work page 2003

  8. [8]

    YuxinGe, MingxiangLi, andZhaoLian,A new invariant on 3-dimensional manifolds and applications, Commun. Contemp. Math.27(2025), no. 5, Paper No. 2450036, 11. MR 4895967

    work page 2025

  9. [9]

    Differential Geom.84 (2010), no

    Yuxin Ge, Chang-Shou Lin, and Guofang Wang,On theσ2-scalar curvature, J. Differential Geom.84 (2010), no. 1, 45–86. MR 2629509

    work page 2010

  10. [10]

    Yuxin Ge and Guofang Wang,On a fully nonlinear Yamabe problem, Ann. Sci. École Norm. Sup. (4) 39(2006), no. 4, 569–598. MR 2290138

    work page 2006

  11. [11]

    ,On a conformal quotient equation, Int. Math. Res. Not. IMRN (2007), no. 6, Art. ID rnm019,

    work page 2007

  12. [12]

    Math.249(2013), 131–160

    ,A new conformal invariant on 3-dimensional manifolds, Adv. Math.249(2013), 131–160. MR 3116569

    work page 2013

  13. [13]

    II, Comm

    ,On a conformal quotient equation. II, Comm. Anal. Geom.21(2013), no. 1, 1–38. MR 3046938

    work page 2013

  14. [14]

    Yuxin Ge, Guofang Wang, and Wei Wei,Optimal geometric inequalities and fully nonlinear conformal flows, preprint (2025)

    work page 2025

  15. [15]

    Yuxin Ge, Guofang Wang, and Chao Xia,On problems related to an inequality of Andrews, De Lellis, and Topping, Int. Math. Res. Not. IMRN (2013), no. 20, 4798–4818. MR 3118877

    work page 2013

  16. [16]

    Z.247(2004), no

    Pengfei Guan, Chang-Shou Lin, and Guofang Wang,Application of the method of moving planes to conformally invariant equations, Math. Z.247(2004), no. 1, 1–19. MR 2054518

    work page 2004

  17. [17]

    Reine Angew

    Pengfei Guan and Guofang Wang,A fully nonlinear conformal flow on locally conformally flat mani- folds, J. Reine Angew. Math.557(2003), 219–238. MR 1978409

    work page 2003

  18. [18]

    J.124(2004), no

    ,Geometric inequalities on locally conformally flat manifolds, Duke Math. J.124(2004), no. 1, 177–212. MR 2072215

    work page 2004

  19. [19]

    Pengfei Guan and Xiangwen Zhang,A class of curvature type equations, Pure Appl. Math. Q.17 (2021), no. 3, 865–907. MR 4278951

    work page 2021

  20. [20]

    Gursky and Jeff A

    Matthew J. Gursky and Jeff A. Viaclovsky,A new variational characterization of three-dimensional space forms, Invent. Math.145(2001), no. 2, 251–278. MR 1872547

    work page 2001

  21. [21]

    Differential Geom.63(2003), no

    ,A fully nonlinear equation on four-manifolds with positive scalar curvature, J. Differential Geom.63(2003), no. 1, 131–154. MR 2015262

    work page 2003

  22. [22]

    ,Prescribing symmetric functions of the eigenvalues of the Ricci tensor, Ann. of Math. (2)166 (2007), no. 2, 475–531. MR 2373147 20 YUXIN GE, GUOF ANG W ANG, AND WEI WEI

    work page 2007

  23. [23]

    1, 45–70

    Gerhard Huisken and Carlo Sinestrari,Convexity estimates for mean curvature flow and singularities of mean convex surfaces, Acta Math.183(1999), no. 1, 45–70. MR 1719551

    work page 1999

  24. [24]

    Mijia Lai,A note on a three-dimensional sphere theorem with integral curvature condition, Commun. Contemp. Math.18(2016), no. 4, 1550070, 8. MR 3493221

    work page 2016

  25. [25]

    Lee and Thomas H

    John M. Lee and Thomas H. Parker,The Yamabe problem, Bull. Amer. Math. Soc. (N.S.)17(1987), no. 1, 37–91. MR 888880

    work page 1987

  26. [26]

    Aobing Li and Yan Yan Li,On some conformally invariant fully nonlinear equations. II. Liouville, Harnack and Yamabe, Acta Math.195(2005), 117–154. MR 2233687

    work page 2005

  27. [27]

    Pure Appl

    Aobing Li and Yanyan Li,On some conformally invariant fully nonlinear equations, Comm. Pure Appl. Math.56(2003), no. 10, 1416–1464. MR 1988895

    work page 2003

  28. [28]

    YanYan Li and Luc Nguyen,A compactness theorem for a fully nonlinear Yamabe problem under a lower Ricci curvature bound, J. Funct. Anal.266(2014), no. 6, 3741–3771. MR 3165241

    work page 2014

  29. [29]

    1365, Springer, Berlin, 1989, pp

    RichardM.Schoen,Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, Topics in calculus of variations (Montecatini Terme, 1987), Lecture Notes in Math., vol. 1365, Springer, Berlin, 1989, pp. 120–154. MR 994021

    work page 1987

  30. [30]

    Differential Geom.77(2007), no

    Wei-MinSheng, NeilS.Trudinger, andXu-JiaWang,The Yamabe problem for higher order curvatures, J. Differential Geom.77(2007), no. 3, 515–553. MR 2362323

    work page 2007

  31. [31]

    Trudinger,Remarks concerning the conformal deformation of Riemannian structures on com- pact manifolds, Ann

    Neil S. Trudinger,Remarks concerning the conformal deformation of Riemannian structures on com- pact manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3)22(1968), 265–274. MR 240748

    work page 1968

  32. [32]

    Trudinger and Xu-Jia Wang,On Harnack inequalities and singularities of admissible metrics in the Yamabe problem, Calc

    Neil S. Trudinger and Xu-Jia Wang,On Harnack inequalities and singularities of admissible metrics in the Yamabe problem, Calc. Var. Partial Differential Equations35(2009), no. 3, 317–338. MR 2481828

    work page 2009

  33. [33]

    Viaclovsky,Conformal geometry, contact geometry, and the calculus of variations, Duke Math

    Jeff A. Viaclovsky,Conformal geometry, contact geometry, and the calculus of variations, Duke Math. J.101(2000), no. 2, 283–316. MR 1738176

    work page 2000

  34. [34]

    J.12(1960), 21–37

    Hidehiko Yamabe,On a deformation of Riemannian structures on compact manifolds, Osaka Math. J.12(1960), 21–37. MR 125546 Institut de Mathématiques de Toulouse,, Université Paul Sabatier,, 118, route de Nar- bonne,, 31062 Toulouse Cedex, France Email address:yge@math.univ-toulouse.fr Albert-Ludwigs-Universität Freiburg, Mathematisches Institut, Eckerstr. 1...

    work page 1960