Completeness of hyperbolic centroaffine hypersurfaces

Abstract

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.