The Bernstein problem for affine maximal hypersurfaces

Research Report MRR99-020

The Bernstein problem for affine maximal hypersurfaces

Neil S. Trudinger, Xu-Jia Wang

Abstract: In this paper, we prove the validity of the Chern conjecture in affine geometry [18], namely that an affine maximal graph of a smooth, locally uniformly convex function on two dimensional Euclidean space, $\R^2$ , must be a paraboloid. More generally, we shall consider the n-dimensional case, $\R^n$ , showing that the corresponding result holds in higher dimensions provided that a uniform, ``strict convexity" condition holds. We 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 $n\ge 10$ .

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