The Bernstein problem for affine maximal hypersurfaces

Date

2000

Authors

Trudinger, Neil
Wang, Xu-Jia

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Volume Title

Publisher

Springer

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, R2, must be a paraboloid. More generally, we shall consider the n-dimensional case, Rn, 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 ≥ 10.

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Source

Inventiones Mathematicae

Type

Journal article

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DOI

Restricted until

2037-12-31