A Bernstein property of affine maximal hypersurfaces
The variational problem for the unimodular-affine area functional leads to a fourth order Euler-Lagrange equation, the vanishing of the affine mean curvature. For locally strongly convex surfaces, the second variation is negative, thus their extremals are ``affine maximal surfaces'' (Calabi). An affine Bernstein problem is the study of such Euler-Lagrange equations under completeness conditions. Proofs for the affine Bernstein conjectures in the versions of Calabi (complete Blaschke metric) and Chern (global graph representation), resp., in dimension \(n=2\), were given in 2000-2001. The authors state and solve an affine Bernstein problem for locally strongly convex hypersurfaces with another complete, affinely invariant metric, namely the metric defined by the Hessian of a graph representation over an \(n\)-dimensional domain \(G \in \mathbb R^{n+1}\); the assumptions imply that the hypersurface is an elliptic paraboloid. The proof works in dimensions \(n=2,3\). The very difficult and deep proof uses a splitting procedure for the solution of the fourth order Euler-Lagrange equation and Hofer's Lemma which is known from symplectic geometry.
scientific article; zbMATH DE number 1925041