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≥10.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Author information
Authors and Affiliations
Additional information
Oblatum 16-IV-1999 & 4-XI-1999¶Published online: 21 February 2000
Rights and permissions
About this article
Cite this article
Trudinger, N., Wang, XJ. The Bernstein problem for affine maximal hypersurfaces. Invent. math. 140, 399–422 (2000). https://doi.org/10.1007/s002220000059
Issue Date:
DOI: https://doi.org/10.1007/s002220000059