Locally convex hypersurfaces of prescribed curvature and boundary
Frmo the introduction: We prove an existence theorem for locally strictly convex hypersurfaces in \(\mathbb{R}^{n+1}\) (or some flat \((n+1)\)-dimensional manifold), prescribed the boundary of the hypersurface and the curvature quotient \(S_{n,l}(\kappa_1,\dots, \kappa_n)\) as a function of the position. Here, we use the notations \(S_{n,l}:= S_n/S_l\), \(S_k\) the order \(k\) elementary symmetric function of \(n\) variables, \(S_0\equiv 1\) and \(\kappa_1,\dots, \kappa_n\) the principle curvatures. The case of the Gauss curvature to \(l=0\) and is included in our discussion. It is one of our goals to restrict the global shape of the hypersurfaces under investigation as little as possible. In particular, they need not be globally convex. More precisely, our surfaces are the image of sections in some smooth but otherwise arbitrary Euclidean straight line bundle. This includes Cartesian and radial graphs as special cases.
scientific article; zbMATH DE number 1234912