Communications in Analysis and Geometry
Volume 24 (2016)
Completeness of hyperbolic centroaffine hypersurfaces
Pages: 59 – 92
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 Kähler manifold of negative scalar curvature.
completeness, centroaffine hypersurfaces, cubic hypersurfaces, projective special real manifolds, special geometry, very special real manifolds, special Kähler manifolds, quaternionic Kähler manifolds, $r$-map, $c$ map
2010 Mathematics Subject Classification