On the geometry of 3-dimensional convex cones

On the interior of an n-dimensional proper convex cone there exists a canonical Riemannian metric, the Cheng-Yau metric. This metric is invariant with respect to linear automorphisms of the cone and decomposes in a direct product of a 1-dimensional radial and a (n-1)-dimensional transversal component. The corresponding family of (n-1)-dimensional submanifolds is the homothetic family of hyperbolic affine spheres which are asymptotic to the boundary of the cone.

For 3-dimensional cones the affine sphere is a Riemann surface which can be uniformized by a global complex coordinate, defined either on the unit disk or the whole complex plane, depending on the conformal type of the surface. In this coordinate, the cubic form of the affine sphere can be represented by a cubic holomorphic differential. This yields a correspondence between holomorphic functions on the disk or on the plane and convex 3-dimensional cones.

We give an overview of existing results on this correspondence and present some new results.


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