Bernstein Problem of Affine Maximal Type Hypersurfaces on Dimension N>=3

TL;DR

This paper constructs non-quadratic affine maximal hypersurfaces in dimensions three and higher, challenging previous conjectures and extending understanding of affine maximal hypersurfaces beyond the two-dimensional case.

Contribution

It provides the first known examples of non-quadratic affine maximal hypersurfaces in dimensions N≥3, expanding the scope of affine geometry research.

Findings

Constructed non-quadratic affine maximal hypersurfaces in N≥3

Extended the class of known affine maximal hypersurfaces beyond quadratic forms

Challenged existing conjectures in affine geometry

Abstract

Bernstein problem for affine maximal type equation has been a core problem in affine geometry. A conjecture proposed firstly by Chern for entire graph and then extended by Trudinger-Wang to its fully generality asserts that any Euclidean complete, affine maximal type, locally uniofrmly convex C^4-hypersurface in R^{N+1} must be an elliptic paraboloid. At the same time, this conjecture was solved completely by Trudinger-Wang for dimension N=2 and \theta=3/4, and later extended by Jia-Li to N=2, \theta\in(3/4,1] (see also [Zhou]). On the past twenty years, much efforts were done toward higher dimensional issues but not really successful yet, even for the case of dimension N=3. In this paper, we will construct non-quadratic affine maximal type hypersurfaces which are Euclidean complete for N>=3, \theta\in(1/2,(N-1)/N).