The σ_k-Yamabe problem revisited

Guofang Wang, Wei Wei, Yuxin Ge

Pith reviewed 2026-05-08 15:44 UTC · model grok-4.3

classification 🧮 math.DG math.AP

keywords σ_k-Yamabe problemconformal geometryscalar curvatureYamabe constantσ2 curvatureclosed manifoldsexistence of metrics

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The pith

If a closed manifold has positive Yamabe constant and positive σ₂-Yamabe constant, then the latter is achieved by a conformal metric.

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

The paper establishes that for a closed manifold with positive Yamabe constant Y1 greater than zero, the σ₂-Yamabe constant defined as an infimum over conformal metrics with positive scalar curvature is attained by some metric in the conformal class, provided this constant is positive. This attainment directly solves the σ₂-Yamabe problem of finding a conformal metric with constant σ₂ curvature. A further consequence is that the infimum remains the same whether or not one requires the σ₂ curvature itself to be positive. These results rely on the positivity conditions and fail in general without the restriction to positive scalar curvature metrics.

Core claim

We prove that on a closed manifold (M, [g0]) with positive Yamabe constant Y1(M, [g0]) > 0, the σ₂-Yamabe constant Y2(M, [g0]) is achieved by a conformal metric g in [g0] assuming Y2(M, [g0]) > 0. This solves the σ₂-Yamabe problem. As a consequence, the infimum over metrics with R_g > 0 equals the infimum over those with σ₂(g) > 0 as well.

What carries the argument

The σ₂-Yamabe constant Y₂(M,[g₀]), defined as the infimum of the normalized integral of σ₂(g) over conformal metrics with positive scalar curvature.

Load-bearing premise

The assumption that the σ₂-Yamabe constant is positive and that the infimum is taken only over metrics with positive scalar curvature; without the latter the result does not hold.

What would settle it

Finding a closed manifold with positive Yamabe constant and positive σ₂-Yamabe constant but where no conformal metric achieves the infimum of the normalized σ₂ integral would falsify the claim.

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.

Desk Editor's Note private letter to a colleague

The paper proves existence of a minimizer for the σ₂-Yamabe constant under Y₁>0 and Y₂>0 with R_g>0, shows the two infima coincide, and gives a counterexample without the scalar curvature restriction.

read the letter

The key takeaway is that on a closed manifold with positive Yamabe constant, if the σ₂-Yamabe constant is also positive then it is achieved by some conformal metric with positive scalar curvature. They further show this infimum equals the one taken only over metrics with positive σ₂, and they exhibit a counterexample where the result fails once the R_g>0 condition is dropped from the definition of Y₂.

Referee Report

0 major / 3 minor

Summary. The paper proves that on a closed manifold (M,[g₀]) with Y₁(M,[g₀])>0, the σ₂-Yamabe constant Y₂(M,[g₀]), defined as the infimum of ∫σ₂(g) dvol(g) / vol(g)^{(n-4)/n} over conformal metrics g∈[g₀] with R_g>0, is achieved by some g∈[g₀] whenever Y₂>0. This yields a solution to the σ₂-Yamabe problem under the stated hypotheses. As a corollary, the infimum over metrics with R_g>0 coincides with the infimum restricted further to those with σ₂(g)>0. The authors also construct counterexamples showing that the conclusions fail if the condition R_g>0 is omitted from the definition of Y₂.

Significance. If the variational existence argument holds, the result supplies a clean resolution of the σ₂-Yamabe problem on manifolds with positive Yamabe constant, together with a sharp counterexample that isolates the necessity of the scalar-curvature positivity constraint. The explicit equality between the two infima and the parameter-free character of the statement are notable strengths.

minor comments (3)

The abstract and introduction should explicitly state the dimension range (presumably n≥5) under which the σ₂ functional is well-defined and the volume-normalization exponent is positive.
In the definition of Y₂, the restriction R_g>0 is imposed only on the admissible metrics; it would be helpful to clarify whether the achieved minimizer automatically satisfies σ₂(g)>0 or whether this follows from the equality of the two infima.
The counterexample section would benefit from a brief remark on whether the constructed metrics can be chosen with constant σ₂ or merely with the infimum not attained.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending minor revision. The referee's summary accurately captures the main results, including the existence theorem for the σ₂-Yamabe problem under the stated hypotheses on Y₁ and Y₂, the equality of the two infima, and the necessity of the R_g > 0 constraint as shown by our counterexamples. No major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; existence result is self-contained

full rationale

The central result is an existence theorem establishing that the infimum defining Y₂(M,[g₀]) is achieved by a conformal metric when Y₁>0, Y₂>0 and the infimum is taken only over metrics with R_g>0. The functional and the constants Y₁, Y₂ are defined directly as infima over admissible conformal classes without reference to the achieving metric itself. The paper supplies both the positive existence statement and an explicit counter-example showing failure when the R_g>0 restriction is removed, confirming that the argument does not reduce to a tautology or to a self-referential fit. No load-bearing step equates the claimed minimizer to its own defining infimum by construction, and no uniqueness theorem or ansatz is imported via self-citation in a way that collapses the derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard background from conformal geometry and elliptic PDE theory; no free parameters, invented entities, or ad-hoc axioms are introduced in the abstract statement.

axioms (2)

domain assumption M is a closed smooth manifold of dimension n
Stated at the beginning of the abstract as the setting for the problem.
domain assumption The Yamabe constant Y₁(M,[g₀]) > 0
Explicit hypothesis required for the achievement result.

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discussion (0)

Reference graph

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