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

For every n≥3n\geq3, let u∈C∞(B2)u\in C^\infty(B_2) be an admissible solution of κ[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. Then sup⁡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.

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Occurred: Sep 2, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: preprint. AI contribution: ai-assisted. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Interior Curvature Estimates for the Graphical Scalar Curvature Equation in All Dimensions · Graphical scalar-curvature interior estimates

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VibeMathed
registry · event recorded by

ChatGPT (OpenAI, model version unstated)
model · ai model contributor · OpenAI

Guohuan Qiu
human · human collaborator

Jin Yan
human · human collaborator

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VibeMathed record: Interior Curvature Estimates for the Graphical Scalar Curvature Equation in All Dimensions evidence for this event

This event attributed to ChatGPT (OpenAI, model version unstated)

For every n≥3n\geq3, let u∈C∞(B2)u\in C^\infty(B_2) be an admissible solution of κ[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. Then sup⁡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. parent of this event

Interior Curvature Estimates for the Graphical Scalar Curvature Equation in All Dimensions parent of this event

This event attributed to Guohuan Qiu

This event attributed to Jin Yan

Interior Curvature Estimates for the Graphical Scalar Curvature Equation in All Dimensions evidence for this event

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