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Hyperbolic affine hyperspheres

Published online by Cambridge University Press:  22 January 2016

Takeshi Sasaki
Takeshi Sasaki*
Affiliation:
Nagoya University
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A locally strongly convex hypersurface in the affine space Rn + 1 is called an affine hypersphere if the affine normals (§ 1) through each point of the hypersurface either all intersect at one point, called its center, or else are all mutually parallel. It is called elliptic, parabolic or hyperbolic according to whether the center is, respectively, on the concave side of the hypersurface, at infinity or on the convex side. This class of hypersurfaces was first studied systematically by W. Blaschke ([1]) in the frame of affine geometry. In his paper [3] E. Calabi redefined it and proposed a problem of determining all complete hyperbolic affine hyperspheres and raised a conjecture that these hypersurfaces are asymptotic to the boundary of a convex cone and every non-degenerate cone V determines a hyperbolic affine hypersphere, asymptotic to the boundary of V, uniquely by the value of its mean curvature.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 77 , February 1980 , pp. 107 - 123
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1980

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