Sharp σ₂ Sobolev trace conjecture

The conjecture asks for the sharp Sobolev trace inequality associated with $\sigma_2$ -curvature on the round hemisphere, together with a classification of its equality cases, equivalently the corresponding smooth critical metrics. The supplied sources do not state the inequality's precise functional, exponent, sharp constant, or equality formulation explicitly; they only identify it as a sharp $\sigma_2$ Sobolev trace conjecture for conformal metrics on the hemisphere.

References

Primary source

arXiv

Additional references

Rigidity, sharp inequalities, and stability for σ₂-curvature — arXiv

differential-geometry fully-nonlinear-elliptic-equations geometric-analysis

Posted 3 weeks ago · edited 3 weeks ago

Progress summary

Refreshed 3 weeks ago

Claimed progress

A September preprint claims an important classification result, but the sharp inequality and its full equality classification remain unproved.

The problem asks for the sharp Sobolev trace inequality associated with $\sigma_2$ -curvature on the round hemisphere and classification of all equality cases, equivalently smooth critical metrics. No proposer or original date is identified. Known partial results include: for $n=4,5$ , local minimizers on the Euclidean ball are the flat metric up to conformal transformations, and the corresponding sharp inequality follows only if minimizers exist.

September 9, 2026: claimed rigidity advance

Wangzhe Wu's preprint Rigidity, sharp inequalities, and stability for $\sigma_2$ -curvature claims that, for $n\geq 5$ , the constant-data case classifies the smooth critical metrics associated with the conjecture and removes a pinching condition in a related hemisphere estimate. The abstract does not establish the complete sharp trace inequality, and the claim has not been independently verified.

Community submission (unverified)

On September 11, 2026, submitted articles describe a variational and Euler–Lagrange framework for $n\geq 5$ and a boundary blow-up, half-space, and Pohozaev analysis. They explicitly present a proof framework rather than a complete resolution, noting that global attainment and the deepest rigidity classification require further hypotheses or established results.

Current status (as of September 2026): The full sharp $\sigma_2$ Sobolev trace conjecture remains open; Wu's claimed rigidity progress and the community submissions are unverified.

Sources

ProofVariational Structure of the σ₂-Sobolev Trace Problem on the Round Hemisphere Conformal Geometry, Admissibility and the Euler–Lagrange System Dr. Arie-Ariadne Dewatson-Ledetsambali Independent Research September 2026See full solutionHide full solution

This article isolates the variational and geometric core of the σ₂-Sobolev trace problem developed in the supplied proof dossier. For n ≥ 5, we consider conformal metrics on the Euclidean ball and formulate the σ₂-functional, its nonlinear boundary curvature, the critical trace exponent and the associated quotient. We derive the structural Euler–Lagrange system used throughout the proof programme and explain the role of the Schouten tensor, the Gårding cone and the first Newton tensor. The article also records the equality model and the precise logical status of the argument: the variational reduction and Euler–Lagrange framework are developed in the dossier, whereas global attainment and the deepest rigidity classification require additional hypotheses or established external results. Keywords: σ₂-curvature; Sobolev trace inequality; Schouten tensor; conformal geometry; Gårding cone; Euler–Lagrange equations; nonlinear elliptic equations. MSC 2020: 35J60; 53C21; 58J05; 35B33.

Article_1_Variational_Sigma2_Sobolev_Trace.pdf336,086 bytesOpen
Article_2_Boundary_Blowup_Pohozaev_Sigma2.pdf337,954 bytesOpen
Article_4_Sharp_Sigma2_Trace_Rigidity_Hemisphere.pdf372,911 bytesOpen
Article_3_Bubble_Decomposition_Energy_Quantization.pdf342,315 bytesOpen

CounterexampleBoundary Blow-Up and the σ₂-Pohozaev Mechanism A Research Framework for Excluding Critical Boundary Concentration Dr. Arie-Ariadne Dewatson-Ledetsambali Independent Research September 2026See full solutionHide full solution

This article develops the boundary blow-up component of the supplied σ₂-Sobolev trace proof dossier. Starting from normalized subcritical minimizing sequences, we formulate the concentration alternative, introduce conformal blow-up coordinates, derive the limiting half-space problem, and explain how the first Newton tensor generates a Pohozaev-type conservation law. The analysis shows why a boundary bubble must be a highly constrained conformal profile and identifies the precise point at which a strict subcritical defect is required. The article is deliberately framed as a proof framework and not as an unsupported claim of a complete resolution. Keywords: boundary blow-up; concentration-compactness; Pohozaev identity; σ₂-curvature; half-space problem; conformal bubble; trace inequality. MSC 2020: 35B33; 35J60; 53C21; 35B44.

sobolev-compressed.pdf1,905,233 bytesOpen