A smooth affine maximal graph in dimension ten
OpenAI's smooth solution (2026) of the affine maximal equation on R^10 with positive definite Hessian that is not a quadratic polynomial. The equation as the Euler–Lagrange equation of the affine area; graphs sqrt(F^2 + t^2) and the reduction to two equations for F and an auxiliary function P; a radial solution through the origin by the contraction mapping principle; and an invariant box for three logarithmic derivatives, whose corner comes from the singular example sqrt(|x|^9 + t^2) of Trudinger and Wang, giving the solution for all radii.
Four lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.
- The affine maximal equation
- Graphs over a convex profile
- A radial solution through the origin
- An invariant box and the entire graph
Theorems and assumptions · Reader and editable sources · Course record
Prerequisites
- Real analysis in one and several variables: the contraction mapping principle, differentiation under the integral sign, Fubini's theorem (core courses Real Analysis I and II).
- Linear algebra: determinants, symmetric matrices and their eigenvalues.