Affine hyperspheres associated to special para-Kahler manifolds

摘要

We prove that any special para-Kahler manifold is intrinsically an improper affine hypersphere. As a corollary, any para-holomorphic function F of n para-complex variables satisfying a non-degeneracy condition defines an improper affine hypersphere, which is the graph of a real function f of 2n variables. We give an explicit formula for the function f in terms of the para-holomorphic function F. Necessary and sufficient conditions for an affine hypersphere to admit the structure of a special para Kahler manifold are given. Finally, it is shown that conical special para-Kahler manifolds are foliated by proper affine hyperspheres of constant mean curvature.