Bernstein problem of affine maximal type hypersurfaces on dimension N ≥ 3

Abstract

Bernstein problem for affine maximal type equation(0.1)uijDijw=0,w≡[det⁡D2u]−θ,∀x∈Ω⊂RN has been a core problem in affine geometry. A conjecture proposed firstly by Chern (1977) [6] for entire graph and then extended by Trudinger-Wang (2000) [14] to its fully generality asserts that any Euclidean complete, affine maximal type, locally uniformly convex C4-hypersurface in RN+1 must be an elliptic paraboloid. At the same time, this conjecture was solved completely by Trudinger-Wang for dimension N=2 and θ=3/4, and later extended by Jia-Li (2009) [12] to N=2,θ∈(3/4,1] (see also Zhou (2012) [16] for a different proof). On the past twenty years, many 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 compete forN≥3,θ∈(1/2,(N−1)/N).