  This paper is concerned with the completeness (with respect to the
centroaffine metric) of hyperbolic centroaffine hypersurfaces which are closed
in the ambient vector space. We show that completeness holds under generic
regularity conditions on the boundary of the convex cone generated by the
hypersurface. The main result is that completeness holds for hyperbolic
components of level sets of homogeneous cubic polynomials. This implies that
every such component defines a complete quaternionic K\"ahler manifold of
negative scalar curvature.
