On Bernstein Theorem of Affine Maximal Equation

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 Maht., 56, 2009, 109-139) to N=3, \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 talk, we will present some known results and new results for the problem.

报告人：杜式忠，汕头大学副教授。主要从事完全非线性偏微分方程与几何分析相关的理论研究，主持国家自然科学基金面上项目一项、青年基金一项。文章发表于Calc. Var. & PDEs., Journal of Differential Equations，Transactions of AMS等数学刊物上。