On Bernstein Problem of Affine Maximal Hypersurfaces

Shi-Zhong Du

Published 2022-10-17Version 1

Bernstein problem for affine maximal type hypersurfaces has been a core problem in affine geometry. A conjecture proposed firstly by Chern (Proc. Japan-United States Sem., Tokyo, 1977, 17-30) for entire graph and then extended by Trudinger-Wang (Invent. Math., 140, 2000, 399-422) to its fully generality asserts that any Euclidean complete, affine maximal type, locally uniformly 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 (Results Math., 56, 2009, 109-139) to N=2, \theta\in(3/4,1] (see also Zhou (Calc. Var. PDEs., 43, 2012, 25-44) for a different proof). 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 firstly solved the Chern-Trudinger-Wang conjecture for affine maximal hypersurfaces on all dimensions N>=3, and then turn to construct non-quadratic affine maximal type hypersurfaces which are Euclidean complete for N>=2, \theta\in(0,(N-1)/N].