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Interior Curvature Estimates for the Graphical Scalar Curvature Equation in All Dimensions

We resolve the long-standing problem of establishing interior C2C^{2} estimates for admissible solutions of the graphical scalar curvature equation in every dimension n≥3n\geq 3. More precisely, we prove interior curvature estimates for admissible solutions to the constant graphical scalar curvature equation. The proof combines Jacobi inequalities with a two-surface maximum principle and a two-surface Pogorelov estimate.

Result
Proved(see note)
Status
Resolved
AI contribution
AI-assisted
Method
Argument
Field
Partial differential equations
Posed by
Interior C^2 estimates for the σ_2 (scalar curvature) equation, open above dimension three since Heinz (n = 2) and Warren–Yuan (n = 3)
Year posed
—
Years open
—
Solved
2026-09-02
Model
ChatGPT (OpenAI, model version unstated)
Vendor
OpenAI
Collaborators
Guohuan Qiu, Jin Yan
Verification
Unreviewed
Publication
Preprint
Significance
30 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

For every n≥3n\geq3, let u∈C∞(B2)u\in C^\infty(B_2) be an admissible solution ofκ[u]∈Γ2,σ2(κ[u])=1, \kappa[u]\in\Gamma_2,\qquad \sigma_2(\kappa[u])=1, with∥u∥L∞(B2)+∥Du∥L∞(B2)≤K. \|u\|_{L^\infty(B_2)}+\|Du\|_{L^\infty(B_2)}\le K. Thensup⁡B1/2∣κ[u]∣≤C(n,K). \sup_{B_{1/2}}|\kappa[u]|\le C(n,K).
Thus an admissible graph of constant scalar curvature with bounded height and slope cannot develop unbounded interior curvature, in any dimension n≥3n\ge3. Prior unrestricted quantitative graphical estimates were known in dimension 33; dimension 44 had only an implicit estimate with extra dependence, and n≥5n\ge5 required additional semiconvexity assumptions.

What the AI did

During development of the uniform-separation and two-surface Pogorelov arguments, the authors used ChatGPT to test candidate comparison and cutoff functions, perform preliminary calculations, and search for counterexamples to proposed differential inequalities. These explorations helped expose the obstruction to one-point comparisons: the two graphical hypersurfaces have mismatched tangent spaces and normals. This informed the replacement of the vertical-gap approach by an ambient-distance comparison and a two-point cutoff. The authors independently checked, corrected, and rewrote all AI-assisted calculations and made the final mathematical decisions.

Verification

Unreviewed. arXiv 2609.02581 (38 pages) read here; the authors write that ChatGPT was used "as an exploratory and computational aid to test candidate" comparison and cutoff functions and search for counterexamples to proposed inequalities, and that they "checked, corrected, and rewrote the AI-assisted calculations" and verified every statement. Author-checked; not refereed; no formalization.

Sources

Submitted by VibeGene on

Changelog2 changes

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