Hypersurfaces with prescribed affine Gauss-Kronecker curvature
It is well known that on a nondegenerate hypersurface in \(\mathbb{R}^{n+1}\), it is possible to introduce a canonical transversal vector field, called the affine normal. Using this affine normal, it is then possible, similar as in the Euclidean case, to introduce a connection (the induced affine connection), a symmetric bilinear form \(h\) (the affine metric) and a 1-1 tensor field \(S\) called the affine shape operator. In the paper under review the authors study hypersurfaces for which \(h(X,Y)\) is positive definite and \(h(SX, Y)\) is negative definite. Such hypersurfaces are called hyperbolic hypersurfaces. The affine Gauss-Kronecker curvature \(S_n\) is defined by \(S_n= (-1)^n\det S\). The authors show that given a bounded convex domain \(\Omega\) and a differentiable function \(\varphi\) on the boundary of \(\Omega\) and a strictly positive function \(S_n\) defined on \(\overline\Omega\) there exists a hyperbolic hypersurface with the following properties: (1) the equiaffine Gauss-Kronecker curvature is the given function \(S_n\), (2) \(M\) is Euclidean complete and also the metric \(h(SX,Y)\) is complete (Weingarten complete) (3) if \(\Omega\) is the ball then \(M\) is also complete with respect to the affine metric. One of the main ideas of the proof is to split the fourth-order PDE into 2 second order PDEs of Monge Ampère type. This was also used in [\textit{A. Li}, \textit{U. Simon} and \textit{B. Chen}, J. Reine Angew. Math. 487, 179-200 (1997; Zbl 0886.35065)].
scientific article; zbMATH DE number 1505833