Results on affine maximal graphs: the equation, the reduction, the radial solution and the invariant box, with their assumptions

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• A smooth nonquadratic affine maximal graph in dimension ten (OpenAI) — Theorem A. There is u in C^infinity(R^10), not a polynomial of degree at most two, with positive definite Hessian at every point, solving the affine maximal equation on R^10. Proved in the last lesson.
• Jacobi's formula — Lemma 1.1. For invertible M, the derivative of det(M + eps N) at eps = 0 is det(M) tr(M^(-1) N).
• Cofactors of a Hessian are divergence free — Lemma 1.2. For u of class C^3 with invertible Hessian, the cofactor matrix U = det(D^2u)(D^2u)^(-1) is symmetric with sum_i d_i U^(ij) = 0.
• First variation of the affine area — Proposition 1.3. For u of class C^4 with positive definite Hessian and compactly supported C^2 variations, the derivative of the affine area is (1/(n+2)) times the integral of phi sum U^(ij) d_ij w; u solves the equation if and only if all these derivatives vanish.
• Hessian of sqrt(F^2 + t^2) — Corollary 1.2. In a suitable basis the Hessian is block diagonal with blocks (F/u) D^2F and F^2/u^3; det D^2u = F^(m+2) u^(-(m+3)) det D^2F, and D^2u > 0 when D^2F > 0.
• Reduction to two equations (OpenAI) — Proposition 2.1. If F, P > 0, D^2F > 0, det D^2F = F P^(-gamma) and tr((D^2F)^(-1) D^2P) + (m+2) P/F = 0, then sqrt(F^2 + t^2) solves the affine maximal equation in m + 1 variables with positive definite Hessian.
• The radial system — Corollary 3.2. For radial F, P with y = F' > 0 and y' > 0, the two equations are equivalent to y^(m-1) y' = r^(m-1) F P^(-gamma) and (y^(m-1) P')' = -(m+2) r^(m-1) P^(1-gamma).
• The fixed point in s = r^2/2 — Proposition 1.3. For small delta the integral equations for (f, p) on [-delta, delta] have a unique solution near (1, 1); it is smooth, with f'(0) = 1 and p'(0) = -(m+2)/m.
• A radial solution through the origin (OpenAI) — Proposition 2.1. F = f(|x|^2/2) and P = p(|x|^2/2) are smooth and positive near 0, with D^2F(0) = I, D^2F > 0, and solve the radial system for 0 < r < epsilon with y, y' > 0 and P' < 0.
• Maximal solutions and continuation — Lemma 1.2. Every initial value problem with a C^1 right side has a maximal solution; if its right end is finite, the solution leaves every compact subset of the domain.
• The autonomous system (OpenAI) — Lemma 3.1. For m = 9, A = ry/F, B = ry'/y and D = -rP'/(PB) satisfy an autonomous polynomial system in log r and tend to (0, 1, 0) as r tends to 0.
• The invariant box (OpenAI) — Lemma 4.1. The box [0, 9/2] x [0, 7/2] x [0, 33/7] is forward invariant for the autonomous system; its upper corner is a zero of the field.
• Existence for all radii (OpenAI) — Proposition 5.1. For m = 9 the radial solution exists for all r > 0 with F, P, F', F'' > 0.

Reading order

Read the lessons in order. The first lesson states the theorem and derives the equation from the affine area; the second reduces it to a system of ordinary differential equations; the third and fourth solve that system near the origin and then for all radii.