Blaschke hypersurfaces with constant negative affine mean curvature (Q2017593)

The authors prove the following main theorems. Theorem 1. Let \(x: M\to\mathbb R^{n+1}\) be a locally strongly convex, affine complete Blaschke hypersurface, satisfying the following conditions: {\parindent=8mm\begin{itemize}\item[(i)] The affine Weingarten operator \(B\) is bounded below: \(B\geq k\cdot \mathrm{id}\) for \(0>k\in R;\) \item[(ii)] \(H=\) const. \(<0\); \item[(iii)] the affine support function \(\Lambda(x_0)\) is bounded for some \(x_o\in R^{n+1}\). \end{itemize}} Then \(x(M)\) is an affine complete hyperbolic affine sphere. Theorem 2. A locally strongly convex affine complete hypersphere, satisfying the following condition \(\chi\geq 0\) on the normalized scalar curvature, is either a quadratic hypersurface or a hyperbolic hypersphere of type \(Q(c,n)\).