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The Bernstein problem for affine maximal hypersurfaces (2000)

Trudinger, Neil S. ; Wang, Xu-Jia

Springer

Inventiones mathematicae 140 (2000), S. 399-422

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ISSN: 1432-1297

Keywords: Mathematics Subject Classification (1991): 53A15, 53A10, 53C45, 35J60

Source: Springer Online Journal Archives 1860-2000

Topics: Mathematics

Notes: 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.

Type of Medium: Electronic Resource

URL: http://dx.doi.org/10.1007/s002220000059

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