Interior Curvature Estimates for the Graphical Scalar Curvature Equation in All Dimensions
resolvedconfidence 70%
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Precise statement
We resolve the long-standing problem of establishing interior $C^{2}$ estimates for admissible solutions of the graphical scalar curvature equation in every dimension $n\geq 3$. 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.
The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement
fidelity, correctness, priority, or novelty.
What AI did
ChatGPT (OpenAI, model version unstated)
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.
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.
For every $n\geq3$, let $u\in C^\infty(B_2)$ be an admissible solution of
$$
\kappa[u]\in\Gamma_2,\qquad \sigma_2(\kappa[u])=1,
$$
with
$$
\|u\|_{L^\infty(B_2)}+\|Du\|_{L^\infty(B_2)}\le K.
$$
Then
$$
\sup_{B_{1/2}}|\kappa[u]|\le C(n,K).
$$
Thus an admissible graph of constant scalar curvature with bounded height and slope cannot develop unbounded interior curvature, in any dimension $n\ge3$. Prior unrestricted quantitative graphical estimates were known in dimension $3$; dimension $4$ had only an implicit estimate with extra dependence, and $n\ge5$ required additional semiconvexity assumptions.
Known method families
argument (source-reported)
Source-reported tools: argument.
Independent: unknown · difference confidence: 0
What remains uncertain
VibeMath has not independently audited the mathematical statement, proof, or novelty claim.
VibeMath has not independently verified the mathematical claim.
AI-attempt independence and training-data exposure are unknown unless explicitly documented.
VibeMath has not independently audited the mathematical statement, proof, or novelty claim.