Affine differential geometry and partial differential equations of fourth order (Q2735667)

The first part of the article is a survey on recent developments of the affine Bernstein problem, the first version of which is the conjecture of S. S. Chern: Let \(x_3=f(x_1,x_2)\) be a strictly convex function defined for all \((x_1,x_2)\in A^2\). If \(M=((x_1,x_2, f(x_1,x_2))\) is an affine maximal surface, then \(M\) must be an elliptic paraboloid. In the case of hypersurfaces in \(A^3\) or \(A^4\) this conjecture was proved by the authors. In higher dimension there is a partial result of A. V. Pogorelov.NEWLINENEWLINENEWLINEThe second version of the affine Bernstein problem is a conjecture of E. Calabi: A locally strongly convex affine-complete surface \(x:M\to A^3\) with \(L_1\equiv 0\), where \(L_i\) is the affine mean curvature, is an elliptic paraboloid. This conjecture is open, but there are partial results by the first author and A. Martinez and F. Milan. The second part of the paper gives a survey on recent results of A.-M. Li, U. Simon, G.-S. Zhao and B.-H. Chen, concerning the following classes of affine hypersurfaces: complete hypersurfaces with \(L_n=\) constant where \(L_n\) is the affine Gauss-Kronecker curvature, and hyperbolic affine hypersurfaces with prescribed affine Gauss-Kronecker curvature. Two results of H.-K. Liao deal with surfaces in three-dimensional affine space with \(L_2=0\).NEWLINENEWLINEFor the entire collection see [Zbl 0954.00038].