Completeness of hyperbolic centroaffine hypersurfaces

TL;DR

This paper investigates the conditions under which hyperbolic centroaffine hypersurfaces are complete with respect to the centroaffine metric, establishing new results for hypersurfaces defined by homogeneous cubic polynomials.

Contribution

It proves that hyperbolic components of level sets of homogeneous cubic polynomials are complete, linking geometric properties to quaternionic Kähler manifolds.

Findings

Completeness holds under generic boundary regularity conditions.

Hyperbolic components of cubic polynomial level sets are complete.

Such hypersurfaces induce complete quaternionic Kähler manifolds.

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.