Interior curvature bounds for graphical sigma₂ equations
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Can one bound interior curvatures of a smooth admissible graph solving σ₂(κ[u])=1 from its height and gradient bounds alone? Here σ₂ is the second elementary symmetric function of principal curvatures, the graph is defined over B₃⁄₂ and dimension n≥3 is arbitrary.
Why it matters and what counts as progress
Why it matters: All-dimensional graphical curvature estimates extend low-dimensional results without importing stronger convexity or gradient-modulus assumptions.
What counts as progress: Prove all-dimensional interior estimates without an extra gradient modulus or extend the equation and admissible-graph class.
How the problem developed
- 2024 · Partial result [1]
Guohuan Qiu Published interior Hessian estimates for a related σ₂ equation in dimension three, under the hypotheses of that paper.
- 2 Sept 2026 · Theorem [2]
Guohuan Qiu and Jin Yan Released an all-dimensional result on interior C² and curvature estimates for admissible solutions of the constant graphical scalar-curvature equation in dimensions n≥3.
- 15 Sept 2026 · Revision [2]
Guohuan Qiu and Jin Yan Revised the manuscript, retaining the all-dimensional theorem and its two-surface maximum-principle and Pogorelov-estimate framework.
AI contribution
Interior Curvature Estimates for the Graphical Scalar Curvature Equation in All Dimensions
Qiu and Yan prove interior curvature bounds in every dimension n≥3 for smooth admissible graphs on B₃⁄₂ solving σ₂(κ[u])=1 with bounded height and gradient. The model-suggested compression inequality in Lemma 3.4 controls a two-surface Jacobi inequality; the model also supplied the auxiliary η in equation 62.
Sources
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Publication dates and editorial changes.
Model-suggested inequality supports graphical curvature estimates