The Bernstein problem for affine maximal hypersurfaces
The Bernstein problem for hypersurfaces with vanishing equiaffine mean curvature (called ``affine minimal'' in the terminology of Chern) was first proposed, for dimension \(n=2\), i.e., for surfaces, in the article [\textit{S. S. Chern}, Proc. Japan-United States Sem., Tokyo, 1977, 17-30 (1979; Zbl 0439.53008)]. Its statement is the following: ``A Euclidean complete, affine minimal, locally strongly convex surface in \(\mathbb{R}^3\) must be an elliptic paraboloid''. Subsequently the problem was considered by Calabi, who showed, first, that if a hypersurface \(M\) is a critical point of the area functional \(A\), the second variation of \(A\) at \(M\) is non-positive, that is, the affine area \(A(M)\) reaches a maximum under smooth interior perturbations. Accordingly he proposed that such an \(M\) be called, more properly, an affine maximal hypersurface. Then, he verified Chern's conjecture under the hypothesis that the affine metric of \(M\) be complete [\textit{E. Calabi}, Am. J. Math. 104, 91-126 (1982; Zbl 0501.53037)]. In the present article, the authors prove the validity of the conjecture in the two-dimensional case. More generally, they consider the \(n\)-dimensional case, showing that the corresponding result holds in higher dimensions provided that a uniform, ``strict convexity'' condition holds. They also extend the notion of ``affine maximal'' to non-smooth convex graphs and produce a counterexample showing that the Bernstein result does not hold in this generality for dimension \(d\geq 10\).
scientific article; zbMATH DE number 1463434