Communications in Analysis and Geometry

Volume 24 (2016)

Number 1

Completeness of hyperbolic centroaffine hypersurfaces

Pages: 59 – 92

DOI: http://dx.doi.org/10.4310/CAG.2016.v24.n1.a3

Authors

V. Cortés (Fachbereich Mathematik und Zentrum für Mathematische Physik, Universität Hamburg, Germany)

M. Nardmann (Fachbereich Mathematik und Zentrum für Mathematische Physik, Universität Hamburg, Germany; and Fakultät für Mathematik, Technische Universität, Dortmund, Germany)

S. Suhr (Fachbereich Mathematik und Zentrum für Mathematische Physik, Universität Hamburg, Germany; Département de Mathématiques et Applications, École Normale Supérieure, Paris; and Université Paris Dauphine, Paris, France)

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 Kähler manifold of negative scalar curvature.

Keywords

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

53A15, 53C26

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Published 6 June 2016